Contents
Preface to the Second Edition
Preface to the First Edition
0 The Origins of Complex Analysis, and Its Challenge to Intuition
0.1 The Origins of Complex Numbers
0.2 The Origins of Complex Analysis
0.3 The Puzzle
0.4 Is Mathematics Discovered or Invented?
0.5 Overview of the Book
1 Algebra of the Complex Plane
1.1 Construction of the Complex Numbers
1.2 The x +iy Notation
1.3 A Geometric Interpretation
1.4 Real and Imaginary Parts
1.5 The Modulus
1.6 The Complex Conjugate
1.7 Polar Coordinates
1.8 The Complex Numbers Cannot be Ordered
1.9 Exercises
2 Topology of the Complex Plane
2.1 Open and Closed Sets
2.2 Limits of Functions
2.3 Continuity
2.4 Paths
2.4.1 Standard Paths
2.4.2 Visualising Paths
2.4.3 The Image of a Path
2.5 Change of Parameter
2.5.1 Preserving Direction
2.6 Subpaths and Sums of Paths
2.7 The Paving Lemma
2.8 Connectedness
2.9 Space-filling Curves
2.10 Exercises
3 Power Series
3.1 Sequences
3.2 Series
3.3 Power Series
3.4 Manipulating Power Series
3.5 Products of Series
3.6 Exercises
4 Differentiation
4.1 Basic Results
4.2 The Cauchy–Riemann Equations
4.3 Connected Sets and Differentiability
4.4 Hybrid Functions
4.5 Power Series
4.6 A Glimpse Into the Future
4.6.1 Real Functions Differentiable Only Finitely Many Times
4.6.2 Bad Behaviour of Real Taylor Series
4.6.3 The Blancmange function
4.6.4 Complex Analysis is Better Behaved
4.7 Exercises
5 The Exponential Function
5.1 The Exponential Function
5.2 Real Exponentials and Logarithms
5.3 Trigonometric Functions
5.4 An Analytic Definition of π
5.5 The Behaviour of Real Trigonometric Functions
5.6 Dynamic Explanation of Euler’s Formula
5.7 Complex Exponential and Trigonometric Functions are Periodic
5.8 Other Trigonometric Functions
5.9 Hyperbolic Functions
5.10 Exercises
6 Integration
6.1 The Real Case
6.2 Complex Integration Along a Smooth Path
6.3 The Length of a Path
6.3.1 Integral Formula for the Length of Smooth Paths and Contours
6.4 If You Took the Short Cut
6.5 Further Properties of Lengths
6.5.1 Lengths of More General Paths
6.6 Regular Paths and Curves
6.6.1 Parametrisation by Arc Length
6.7 Regular and Singular Points
6.8 Contour Integration
6.8.1 Definition of Contour Integral
6.9 The Fundamental Theorem of Contour Integration
6.10 An Integral that Depends on the Path
6.11 The Gamma Function
6.11.1 Known Properties of the Gamma Function
6.12 The Estimation Lemma
6.13 Consequences of the Fundamental Theorem
6.14 Exercises
7 Angles, Logarithms, and the Winding Number
7.2 The Argument of a Complex Number
7.3 The Complex Logarithm
7.4 The Winding Number
7.5 The Winding Number as an Integral
7.6 The Winding Number Round an Arbitrary Point
7.7 Components of the Complement of a Path
7.8 Computing the Winding Number by Eye
7.9 Exercises
8 Cauchy’s Theorem
8.1 The Cauchy Theorem for a Triangle
8.2 Existence of an Antiderivative in a Star Domain
8.3 An Example – the Logarithm
8.4 Local Existence of an Antiderivative
8.5 Cauchy’s Theorem
8.6 Applications of Cauchy’s Theorem
8.6.1 Cuts and Jordan Contours
8.7 Simply Connected Domains
8.8 Exercises
9 Homotopy Versions of Cauchy’s Theorem
9.1 Informal Description of Homotopy
9.2 Integration Along Arbitrary Paths
9.3 The Cauchy Theorem for a Boundary
9.4 Formal Definition of Homotopy
9.5 Fixed End Point Homotopy
9.6 Closed Path Homotopy
9.7 Converse to Cauchy’s Theorem
9.8 The Cauchy Theorems Compared
9.9 Exercises
10 Taylor Series
10.1 Cauchy Integral Formula
10.2 Taylor Series
10.3 Morera’s Theorem
10.4 Cauchy’s Estimate
10.5 Zeros
10.6 Extension Functions
10.7 Local Maxima and Minima
10.8 The Maximum Modulus Theorem
10.9 Exercises
11 Laurent Series
11.1 Series Involving Negative Powers
11.2 Isolated Singularities
11.3 Behaviour Near an Isolated Singularity
11.4 The Extended Complex Plane, or Riemann Sphere
11.5 Behaviour of a Differentiable Function at Infinity
11.6 Meromorphic Functions
11.7 Exercises
12 Residues
12.1 Cauchy’s Residue Theorem
12.2 Calculating Residues
12.3 Evaluation of Definite Integrals
12.4 Summation of Series
12.5 Counting Zeros
12.6 Exercises
13 Conformal Transformations
13.1 Measurement of Angles
13.1.1 Real Numbers Modulo 2π
13.1.2 Geometry of R/2π
13.1.3 Operations on Angles
13.1.4 The Argument Modulo 2π
13.2 Conformal Transformations
13.3 Critical Points
13.4 Möbius Maps
13.4.1 Möbius Maps Preserve Circles
13.4.2 Classification of Möbius Maps
13.4.3 Extension of Möbius Maps to the Riemann Sphere
13.5 Potential Theory
13.5.1 Laplace’s Equation
13.5.2 Design of Aerofoils
13.6 Exercises
14 Analytic Continuation
14.1 The Limitations of Power Series
14.2 Comparing Power Series
14.3 Analytic Continuation
14.3.1 Direct Analytic Continuation
14.3.2 Indirect Analytic Continuation
14.3.3 Complete Analytic Functions
14.4 Multiform Functions
14.4.1 The Logarithm as a Multiform Function
14.4.2 Singularities
14.5 Riemann Surfaces
14.5.1 Riemann Surface for the Logarithm
14.5.2 Riemann Surface for the Square Root
14.5.3 Constructing a General Riemann Surface by Gluing
14.6 Complex Powers
14.7 Conformal Maps Using Multiform Functions
14.8 Contour Integration of Multiform Functions
14.9 Exercises
15 Infinitesimals in Real and Complex Analysis
15.1 Infinitesimals
15.2 The Relationship Between Real and Complex Analysis
15.2.1 Critical Points
15.3 Interpreting Power Series Tending to Zero as Infinitesimals
15.4 Real Infinitesimals as Variable Points on a Number Line
15.5 Infinitesimals as Elements of an Ordered Field
15.6 Structure Theorem for any Ordered Extension Field of R
15.7 Visualising Infinitesimals as Points on a Number Line
15.8 Complex Infinitesimals
15.9 Non-standard Analysis and Hyperreals
15.10 Outline of the Construction of Hyperreal Numbers
15.11 Hypercomplex Numbers
15.12 The Evolution of Meaning in Real and Complex Analysis
15.12.1 A Brief History
15.12.2 Non-standard Analysis in Mathematics Education
15.12.3 Human Visual Senses
15.12.4 Computer Graphics
15.12.5 Summary
15.13 Exercises
16 Homology Version of Cauchy’s Theorem
16.0.1 Outline of Chapter
16.0.2 Group-theoretic Interpretation
16.1 Chains
16.2 Cycles
16.2.1 Sums and Formal Sums of Paths
16.3 Boundaries
16.4 Homology
16.5 Proof of Cauchy’s Theorem, Homology Version
16.5.1 Grid of Rectangles
16.5.2 Proof of Theorem 16.2
16.5.3 Rerouting Segments
16.5.4 Resumption of Proof of Theorem 16.2
16.6 Cauchy’s Residue Theorem, Homology Version
16.7 Exercises
17 The Road Goes Ever On
17.1 The Riemann Hypothesis
17.2 Modular Functions
17.3 Several Complex Variables
17.4 Complex Manifolds
17.5 Complex Dynamics
17.6 Epilogue
References
Index

##### Citation preview

Complex Analysis Second edition This new edition of a classic textbook develops complex analysis from the established theory of real analysis by emphasising the differences that arise as a result of the richer geometry of the complex plane. Key features of the authors’ approach are to use simple topological ideas to translate visual intuition into rigorous proof, and, in this edition, to address the conceptual conﬂicts between pure and applied approaches head-on. Beyond the material of the clariﬁed and corrected original edition, there are three new chapters: Chapter 15 on inﬁnitesimals in real and complex analysis; Chapter 16 on homology versions of Cauchy’s Theorem and Cauchy’s Residue Theorem, linking back to geometric intuition; and Chapter 17 outlines some more advanced directions in which complex analysis has developed, and continues to evolve into the future. With numerous worked examples and exercises, clear and direct proofs, and a view to the future of the subject, this is an invaluable companion for any modern complex analysis course. Ian Stewart, FRS is Emeritus Professor of Mathematics at the University of Warwick. He is author or coauthor of over 190 research papers and is the bestselling author of over 120 books, from research monographs and textbooks to popular science and science ﬁction. His awards include the Royal Society’s Faraday Medal, the IMA Gold Medal, the AAAS Public Understanding of Science Award, the LMS/IMA Zeeman Medal, the Lewis Thomas Prize, and the Euler Book Prize. He is an honorary wizard of the Discworld’s Unseen University. David Tall is Emeritus Professor of Mathematical Thinking at the University of Warwick and is known internationally for his contributions to mathematics education. He is author or coauthor of over 200 papers and 40 books and educational computer software, covering all levels from early childhood to research mathematics.

Complex Analysis (The Hitch Hiker’s Guide to the Plane) Second edition I A N S T E WA RT D AV I D TA L L University of Warwick

University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 79 Anson Road, #06–04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781108436793 DOI: 10.1017/9781108505468 First edition ± c Cambridge University Press 1983 Second edition ± c JOAT Enterprises and David Tall 2018

This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1983 Thirteenth reprint 2004 Second edition 2018 Printed in the United Kingdom by TJ International Ltd, Padstow, Cornwall 2018 A catalogue record for this publication is available from the British Library. Library of Congress Cataloging-in-Publication Data Names: Stewart, Ian, 1945– author. | Tall, David Orme, author. Title: Complex analysis : the hitch hiker’s guide to the plane / Ian Stewart (University of Warwick), David Tall (University of Warwick). Other titles: Hitch hiker’s guide to the plane Description: Second edition. | Cambridge : Cambridge University Press, 2018. | Includes bibliographical references and index. Identiﬁers: LCCN 2018007009 | ISBN 9781108436793 (alk. paper) Subjects: LCSH: Functions of complex variables. | Numbers, Complex. | Geometry, Analytic–Plane. Classiﬁcation: LCC QA331 .S85 2018 | DDC 515/.9–dc23 LC record available at https://lccn.loc.gov/2018007009 ISBN 978-1-108-43679-3 Paperback Additional resources for this publication at www.cambridge.org/Stewart&Tall2ed Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Contents

Preface to the Second Edition Preface to the First Edition

page xi xiv

0

The Origins of Complex Analysis, and Its Challenge to Intuition 0.1 The Origins of Complex Numbers 0.2 The Origins of Complex Analysis 0.3 The Puzzle 0.4 Is Mathematics Discovered or Invented? 0.5 Overview of the Book

1 1 5 6 7 10

1

Algebra of the Complex Plane 1.1 Construction of the Complex Numbers 1.2 The x iy Notation 1.3 A Geometric Interpretation 1.4 Real and Imaginary Parts 1.5 The Modulus 1.6 The Complex Conjugate 1.7 Polar Coordinates 1.8 The Complex Numbers Cannot be Ordered 1.9 Exercises

13 13 15 16 17 17 18 19 20 21

Topology of the Complex Plane 2.1 Open and Closed Sets 2.2 Limits of Functions 2.3 Continuity 2.4 Paths 2.4.1 Standard Paths 2.4.2 Visualising Paths 2.4.3 The Image of a Path 2.5 Change of Parameter 2.5.1 Preserving Direction 2.6 Subpaths and Sums of Paths 2.7 The Paving Lemma

24 26 27 30 35 35 37 37 38 39 39 43

+

2

vi

Contents

2.8 2.9 2.10

Connectedness Space-ﬁlling Curves Exercises

46 52 55

3

Power Series 3.1 Sequences 3.2 Series 3.3 Power Series 3.4 Manipulating Power Series 3.5 Products of Series 3.6 Exercises

59 59 63 66 69 71 72

4

Differentiation 4.1 Basic Results 4.2 The Cauchy–Riemann Equations 4.3 Connected Sets and Differentiability 4.4 Hybrid Functions 4.5 Power Series 4.6 A Glimpse Into the Future 4.6.1 Real Functions Differentiable Only Finitely Many Times 4.6.2 Bad Behaviour of Real Taylor Series 4.6.3 The Blancmange function 4.6.4 Complex Analysis is Better Behaved 4.7 Exercises

75 75 78 82 83 84 87 87 88 89 91 92

5

The Exponential Function 5.1 The Exponential Function 5.2 Real Exponentials and Logarithms 5.3 Trigonometric Functions 5.4 An Analytic Deﬁnition of 5.5 The Behaviour of Real Trigonometric Functions 5.6 Dynamic Explanation of Euler’s Formula 5.7 Complex Exponential and Trigonometric Functions are Periodic 5.8 Other Trigonometric Functions 5.9 Hyperbolic Functions 5.10 Exercises

96 96 98 99 100 101 103 104 105 106 107

Integration 6.1 The Real Case 6.2 Complex Integration Along a Smooth Path 6.3 The Length of a Path 6.3.1 Integral Formula for the Length of Smooth Paths and Contours 6.4 If You Took the Short Cut 6.5 Further Properties of Lengths

111 112 113 117 119 122 122

π

6

...

Contents

6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14

6.5.1 Lengths of More General Paths Regular Paths and Curves 6.6.1 Parametrisation by Arc Length Regular and Singular Points Contour Integration 6.8.1 Deﬁnition of Contour Integral The Fundamental Theorem of Contour Integration An Integral that Depends on the Path The Gamma Function 6.11.1 Known Properties of the Gamma Function The Estimation Lemma Consequences of the Fundamental Theorem Exercises

vii

123 124 126 127 130 131 133 136 137 139 140 143 146

7

Angles, Logarithms, and the Winding Number 7.1 Radian Measure of Angles 7.2 The Argument of a Complex Number 7.3 The Complex Logarithm 7.4 The Winding Number 7.5 The Winding Number as an Integral 7.6 The Winding Number Round an Arbitrary Point 7.7 Components of the Complement of a Path 7.8 Computing the Winding Number by Eye 7.9 Exercises

149 150 151 153 155 159 159 160 161 164

8

Cauchy’s Theorem 8.1 The Cauchy Theorem for a Triangle 8.2 Existence of an Antiderivative in a Star Domain 8.3 An Example – the Logarithm 8.4 Local Existence of an Antiderivative 8.5 Cauchy’s Theorem 8.6 Applications of Cauchy’s Theorem 8.6.1 Cuts and Jordan Contours 8.7 Simply Connected Domains 8.8 Exercises

169 171 173 175 176 177 180 181 183 184

9

Homotopy Versions of Cauchy’s Theorem 9.1 Informal Description of Homotopy 9.2 Integration Along Arbitrary Paths 9.3 The Cauchy Theorem for a Boundary 9.4 Formal Deﬁnition of Homotopy 9.5 Fixed End Point Homotopy 9.6 Closed Path Homotopy 9.7 Converse to Cauchy’s Theorem

187 187 189 191 195 197 198 201

viii

Contents

9.8 9.9

The Cauchy Theorems Compared Exercises

202 204

10

Taylor Series 10.1 Cauchy Integral Formula 10.2 Taylor Series 10.3 Morera’s Theorem 10.4 Cauchy’s Estimate 10.5 Zeros 10.6 Extension Functions 10.7 Local Maxima and Minima 10.8 The Maximum Modulus Theorem 10.9 Exercises

207 208 209 212 213 214 217 219 220 221

11

Laurent Series 11.1 Series Involving Negative Powers 11.2 Isolated Singularities 11.3 Behaviour Near an Isolated Singularity 11.4 The Extended Complex Plane, or Riemann Sphere 11.5 Behaviour of a Differentiable Function at Inﬁnity 11.6 Meromorphic Functions 11.7 Exercises

225 225 230 232 234 236 237 239

12

Residues 12.1 Cauchy’s Residue Theorem 12.2 Calculating Residues 12.3 Evaluation of Deﬁnite Integrals 12.4 Summation of Series 12.5 Counting Zeros 12.6 Exercises

243 243 246 248 258 261 263

13

Conformal Transformations 13.1 Measurement of Angles 13.1.1 Real Numbers Modulo 2 13.1.2 Geometry of R 2 13.1.3 Operations on Angles 13.1.4 The Argument Modulo 2 13.2 Conformal Transformations 13.3 Critical Points 13.4 Möbius Maps 13.4.1 Möbius Maps Preserve Circles 13.4.2 Classiﬁcation of Möbius Maps 13.4.3 Extension of Möbius Maps to the Riemann Sphere 13.5 Potential Theory

268 268 268 269 270 270 271 276 278 278 279 281 281

π

π

Contents

13.6

13.5.1 Laplace’s Equation 13.5.2 Design of Aerofoils Exercises

ix

281 283 284

14

Analytic Continuation 14.1 The Limitations of Power Series 14.2 Comparing Power Series 14.3 Analytic Continuation 14.3.1 Direct Analytic Continuation 14.3.2 Indirect Analytic Continuation 14.3.3 Complete Analytic Functions 14.4 Multiform Functions 14.4.1 The Logarithm as a Multiform Function 14.4.2 Singularities 14.5 Riemann Surfaces 14.5.1 Riemann Surface for the Logarithm 14.5.2 Riemann Surface for the Square Root 14.5.3 Constructing a General Riemann Surface by Gluing 14.6 Complex Powers 14.7 Conformal Maps Using Multiform Functions 14.8 Contour Integration of Multiform Functions 14.9 Exercises

289 289 291 293 293 295 296 296 297 298 299 299 300 301 302 304 305 311

15

Inﬁnitesimals in Real and Complex Analysis 15.1 Inﬁnitesimals 15.2 The Relationship Between Real and Complex Analysis 15.2.1 Critical Points 15.3 Interpreting Power Series Tending to Zero as Inﬁnitesimals 15.4 Real Inﬁnitesimals as Variable Points on a Number Line 15.5 Inﬁnitesimals as Elements of an Ordered Field 15.6 Structure Theorem for any Ordered Extension Field of R 15.7 Visualising Inﬁnitesimals as Points on a Number Line 15.8 Complex Inﬁnitesimals 15.9 Non-standard Analysis and Hyperreals 15.10 Outline of the Construction of Hyperreal Numbers 15.11 Hypercomplex Numbers 15.12 The Evolution of Meaning in Real and Complex Analysis 15.12.1 A Brief History 15.12.2 Non-standard Analysis in Mathematics Education 15.12.3 Human Visual Senses 15.12.4 Computer Graphics 15.12.5 Summary 15.13 Exercises

315 316 318 320 322 323 324 327 328 331 333 336 337 341 341 342 344 345 346 346

x

Contents

16

Homology Version of Cauchy’s Theorem 16.0.1 Outline of Chapter 16.0.2 Group-theoretic Interpretation 16.1 Chains 16.2 Cycles 16.2.1 Sums and Formal Sums of Paths 16.3 Boundaries 16.4 Homology 16.5 Proof of Cauchy’s Theorem, Homology Version 16.5.1 Grid of Rectangles 16.5.2 Proof of Theorem 16.2 16.5.3 Rerouting Segments 16.5.4 Resumption of Proof of Theorem 16.2 16.6 Cauchy’s Residue Theorem, Homology Version 16.7 Exercises

17

The Road Goes Ever On 17.1 The Riemann Hypothesis 17.2 Modular Functions 17.3 Several Complex Variables 17.4 Complex Manifolds 17.5 Complex Dynamics 17.6 Epilogue

374 374 377 378 379 379 381

References Index

382 383

...

350 351 353 354 356 357 358 360 362 363 365 366 368 368 370

Preface to the Second Edition

The ﬁrst edition of Complex Analysis focused on generalising concepts from real analysis to the complex case. Where there were differences, we looked at the geometric picture to see why they were happening. This second edition does the same, but it also focuses on the increasing sophistication of mathematical ideas as we build from intuition to rigour, in a manner where greater understanding leads to more sophisticated intuitions and ways of working. New concepts and methods often start out in a technical way, with problematic aspects that conﬂict with intuition. As well as generalising real analysis, we move beyond it by addressing these conceptual conﬂicts, resolving them, and providing more sophisticated concepts and methods appropriate to complex analytic functions. This approach is used throughout the book. So, for example, the text now includes a short (but complete) discussion of the construction of a space-ﬁlling curve, to challenge our intuition about continuity and to explain why we have had to be careful with topological assertions that appear obvious. The treatment here is simpler than most of the literature on space-ﬁlling curves. We have spent some time examining different notions of a path, especially the role of smoothness. We have added three new chapters. Chapter 15 introduces ideas about inﬁnitesimals in real and complex analysis, thought of as variables that tend to zero, and formulated as elements of extensions of the real and complex ﬁelds. Chapter 16 gives a formal link from analysis back to geometric intuition, formulating and proving homology versions of Cauchy’s Theorem and Cauchy’s Residue Theorem. Chapter 17 outlines a few of the more advanced directions in which complex analysis has developed, and continues to evolve into the future. Chapter 15 has been added for the following reasons. Since the ﬁrst edition appeared in 1983, the ways in which we operate mathematically have changed dramatically. Not only are there computers that perform numerical and symbolic operations at a speed way beyond that previously available to the individual mind; there are also interactive graphics drawn on high-resolution screens that let us visualise mathematical ideas in completely new ways. In particular, we can dynamically magnify pictures to see tiny detail that lets us represent ‘arbitrarily small’ quantities. This second edition therefore includes an extra chapter to introduce formally deﬁned inﬁnitesimals that lie in an ordered extension ﬁeld K of the real numbers, which can be manipulated algebraically and visualised formally on an extended number line. This approach generalises to the complex case using the ﬁeld K(i) where i2 = −1, which

xii

Preface to the Second Edition

can be visualised in the extended complex plane. This construction offers a meaningful bridge between the epsilon-delta rigour of pure mathematics and the intuitive use of inﬁnitesimals in applications. It can easily be shown that any proper ordered ﬁeld extension K of the reals must contain inﬁnitesimal elements x: that is, elements that are not zero yet satisfy |x| < r for all positive real numbers r. Using the completeness of the real numbers, we prove a simple theorem that any ﬁnite element of K has the form k = c + h, where c is real and h is inﬁnitesimal or zero. A transformation in the form m(x) = (x − c)/ε , where ε is a positive inﬁnitesimal, then lets us magnify inﬁnitesimal detail near c and see it with our unaided human eyes in a real picture. This technique extends to the complex case in the ﬁeld K(i). We can now illustrate why complex analysis is so different from real analysis. A differentiable complex function deﬁned on an open set is locally expressible as a power series, and we may take K to be the smallest ordered extension ﬁeld generated by a ∑ single inﬁnitesimal ε . The elements are power series r ≥n ar εr in ε with possibly a ﬁnite number of terms in 1/ε , and each non-zero element has an order of inﬁnitesimality n related to the ﬁrst non-zero coefﬁcient an (where the element may be inﬁnite if n is negative). Meanwhile a differentiable real function may be differentiable once but not twice, and this requires a much more sophisticated extension ﬁeld K such as that given by the logical theory of non-standard analysis. While Gottfried Leibniz imagined inﬁnitesimals of different orders, non-standard analysis fails to have this property and requires a much more sophisticated construction. At the end of the chapter we compare and contrast the various theories within a single framework. Chapter 16 on homology complements Chapter 9 on homotopy versions of Cauchy’s Theorem, and logically it could have been placed immediately after that. We postpone it to the penultimate chapter because we do not wish to delay the more practical payoff from Cauchy’s Theorem – Taylor and Laurent series, residues, evaluation of integrals, summation of series, and so on. Homology can be thought of as a way of characterising ‘holes’ in a topological space, which here is the domain of a complex function f . Singularities, where f is not differentiable, create such holes, and homology helps to describe the topological effect of singularities; for example, in the homology version of Cauchy’s Residue Theorem. To avoid including big chunks of algebraic topology, our approach to homology is based on step paths in open subsets of the plane, one of the main simplifying tools in this book. The proof is ‘bare hands’ and exploits the simple geometry of step paths and the abelian group structure of homology. Chapter 17 has been included to make it clear that complex analysis is still a major area of mathematical research. Complete though the classical theory may seem to be, there are numerous generalisations and new questions. The main topics mentioned are the Riemann Hypothesis, modular functions, several complex variables, complex manifolds, and complex dynamics – leading to the fractal geometry of Julia sets and the famous Mandelbrot set. In this new edition of Complex Analysis we have corrected all known typographical errors, simplifed some proofs, and reorganised the material in mostly harmless ways to

Preface to the Second Edition

xiii

improve readability. We have brought the text and layout into line with current practice, and redrawn all the ﬁgures. Proofs, deﬁnitions, and examples are terminated with the ‘end of proof’ symbol ±. The same symbol indicates the absence of a proof when the result is clear or has already been proved. Contrary to the prevailing wisdom, we do not insert punctuation marks at the end of displayed formulas. (Your tutors may object to this. Tradition is on their side. If they do, they can set you an extra exercise: insert all missing punctuation.) But it is now the twenty-ﬁrst century. No one puts full stops (US: periods) at the end of book titles, or chapter or section headings. So why do this in displayed formulas, where it may cause confusion because punctuation marks are also often part of the symbolism? We suggest that clean typography should override pedantic punctuation. Formulas in the main text are another matter; here the absence of punctuation can cause confusion. We have followed tradition here.

Online Supplementary Material Supplementary material including a concordance showing in more detail the changes between the previous edition and this one, and links to GeoGebra, can be found on the Cambridge University Press website: www.cambridge.org/Stewart&Tall2ed.

Preface to the First Edition

Students faced with a course on ‘Complex Analysis’ often ﬁnd it to be just that – complex. In the sense of ‘complicated’. It’s true, of course, that the proofs of some of the major theorems in the subject can demand a certain technical versatility. But in many ways, on a conceptual level, complex analysis is actually easier than real analysis; it just isn’t always taught that way. This book is intended for use at the level of second or third year undergraduates, and it is based on experience accumulated from teaching such courses over the past decade. To exhibit the inherent simplicity of complex analysis we have organised the material around two basic principles: (1) generalise from the real case, and (2) when that reveals new phenomena, use the rich geometry of the plane to understand them. Our aim throughout is to encourage geometric thinking, with the proviso that it must be adequately backed by analytic rigour. The opening chapter sets the work in its historical context, and the history is often alluded to later as partial motivation. However, we feel that cultural changes often affect the status of conceptual problems: what was once an important difﬁculty can become a triviality when viewed with hindsight. It is not always necessary to drag today’s students through yesterday’s hang-ups. We argue the point at greater length below: it is fundamental to our entire approach.

0

The Origins of Complex Analysis, and Its Challenge to Intuition

In a lecture in 1886, Leopold Kronecker asserted that the integers are made by God and all the rest is the work of Man (Gray [7]). If so, complex numbers are certainly one of humanity’s most intriguing mathematical artefacts. For centuries they have been a wonder to mathematicians and philosophers alike. It took nearly 300 years from their ﬁrst appearance in Girolamo Cardano’s Ars Magna (The Great Art) to the publication of a formal deﬁnition that satisﬁes modern standards of rigour. Building on such foundations, the initiated reader might be forgiven for thinking that complex analysis must be an incredibly complicated theory. Yet here we come to a historical puzzle. Although it took nearly three centuries to obtain a satisfactory treatment of complex numbers, it then took less than a tenth of that time to complete a major part of complex analysis, which is far more sophisticated and extensive. Obviously the numbers must come ﬁrst, or there is nothing to do analysis with, but the timescale is surprising. A possible explanation is that setting up the foundations adequately involved deep problems of a philosophical nature: it took a long time to come to grips with them, but once the ‘breakthrough’ had occurred, the further development was easy by comparison. History suggests otherwise.

0.1

The Origins of Complex Numbers Cardano’s celebrated Ars Magna of 1545 is one of the most important early algebra texts. Diophantus’s Arithmetica of about 250 discussed the solution of equations and introduced a rudimentary form of algebraic notation. Muhammad al-Khwarizmi’s Al-kitab al-mukhtasar ﬁ hisab al-gabr wa’l-muqabala (The Compendious Book on Calculation by Completion and Balancing) appeared around 820. Its translation into Latin as Liber Algebrae et Almucabola gave us the word ‘algebra’. Al-Khwarizmi’s discussion was verbal, with no symbols but occasional diagrams. Cardano introduced a systematic algebraic notation, very different from what we use today. He used this to present the newly discovered solutions of cubic and quartic equations. His book contained the solution of cubics discovered by Scipione del Ferro around 1500, and independently by Niccolo Fontana (nicknamed ‘Tartaglia’, the stammerer) around 1535. The high point of the text is the solution of quartic equations found by Cardano’s student Lodovico Ferrari. The tangled tale of alleged duplicity and public

2

The Origins of Complex Analysis, and Its Challenge to Intuition

controversy that accompanied these discoveries can be found in Stewart [19, 20] and other historical sources. Ars Magna also discussed the simultaneous equations x + y = 10 xy = 40

and obtained a solution (in modern notation) of the form x=5+

−15

y=5−

−15

Cardano gave no interpretation for the square root of a negative number, but he did observe that, on the assumption that the quantities obey the usual algebraic rules, we can check that they satisfy the equations. His attitude to the discovery was dismissive: ‘So progresses arithmetic subtlety, the end of which . . . is as reﬁned as it is useless.’ In the same book he observed that applying Tartaglia’s formula to the cubic equation x3 = 15x + 4 leads to the solution

±

(0.1)

± √ √ x = 2 + −121 + 2 − −121 in contrast to the obvious answer x = 4. 3

3

In both instances there was a conﬂict between the intuition about numbers that mathematicians had built up over the years, and the formal behaviour of the symbolic manipulations that Cardano was carrying out. It took centuries for mathematicians to extend the number concept and develop a reﬁned intuition in which Cardano’s observations make sense. The ﬁrst step happened not long after, however. Raphael Bombelli (1526–73) suggested a way to reconcile the two solutions of (0.1) by manipulating the ‘impossible’ roots as if they are ordinary numbers. Since (2 ±

Cardano’s expression becomes x = (2 +

√ −1)3 = 2 ± −121

√ −1) + (2 − −1) = 4

and the ‘impossible’ root is just the familiar root in a complex disguise. Bombelli’s work was the ﬁrst hint that complex numbers can prove useful in solving real mathematical problems. But the message took a long time to sink in. In La Géometrie (1637), René Descartes made the distinction between ‘real’ and ‘imaginary’ numbers, interpreting the occurrence of imaginaries as a sign that the problem concerned is insoluble, an opinion shared by Isaac Newton at a later date. However, this view sits uneasily with Bombelli’s realisation that a formula involving complex numbers sometimes leads to a real solution, suggesting that the issue is not that simple. John Wallis [25] represented a complex number geometrically in his Algebra of 1685. On a ﬁxed line the real part of the number was measured off (in the direction given by its sign); then the imaginary part was measured off at right angles, Figure 1. But this idea was largely forgotten.

0.1 The Origins of Complex Numbers

3

Figure 1 Wallis’s geometric representation of a complex number.

In 1702 John Bernoulli was evaluating integrals of the form

²

ax2

dx + bx + c

by partial fractions. Using the philosophy that complex numbers can be manipulated like real ones, he wrote the integrand as ax 2

1 + bx + c

= x −A α + x −B β

(using modern notation) where α, β are the roots of the quadratic denominator, and obtained the integral in the form A log(x − α) + B log(x − β ) His bold decision to use the same method when the quadratic had no real solutions led to logarithms of complex numbers. But what were they? Both Bernoulli and Leibniz used the method, and by 1712 they were engaged in controversy. Leibniz asserted that the logarithm of a negative number is complex, while Bernoulli insisted it is real. Bernoulli argued that, since d(−x) dx = −x x

it follows by integration that log( −x) = log(x). Leibniz, on the other hand, insisted that the integration was correct only for positive x. Once again, formal calculations that seemed sensible were in conﬂict with intuition. Leonhard Euler resolved the controversy in favour of Leibniz in 1749, pointing out that integration requires an arbitrary constant log(−x) = log(x) + c

a point that Bernoulli had ignored. By formally manipulating expressions involving complex numbers, Euler derived a host of theoretical relations, including the famous formula of 1748: eiθ

Putting θ

= π we ﬁnd

= cos θ + i sin θ ei π

= −1

(0.2)

(0.3)

4

The Origins of Complex Analysis, and Its Challenge to Intuition

a fantastic relation that blends the three mathematical symbols e, i, and π in one surprising equation. The formula (0.3) is widely referred to as Euler’s formula, although he never published it explicitly. He did publish (0.2), of which it is a simple corollary, and this is also known as Euler’s formula. However, a formula equivalent to (0.2) had been found earlier by Roger Cotes in 1714. Extending the theory of logarithms to the complex case by deﬁning log z = w if and only if ew

=z

we obtain other intriguing results. Formal manipulation gives elog z +mπ i

= elog z(eπ i )m = z · (−1)m

For an even integer m = 2n this gives

elog z+2nπ i

=z

So log z + 2nπ i is also a logarithm of z: the complex logarithm is many-valued. For an odd integer m = 2n + 1 we have elog z+(2n+1)π i

whence

= −z

log( −z) = log z + (2n + 1)π i

This resolves the Leibniz–Bernoulli controversy: if x is real and positive, then log(−x) must be complex. As mathematicians reﬁned their intuition to encompass complex numbers, everything started to ﬁt together and make sense. The theory of complex numbers grew ever more fascinating. What was lacking was an interpretation that explained precisely what these entities are – a formal counterpart to the newly extended intuitions. In 1797 Caspar Wessel published a paper in Danish describing the representation of a complex number as a point in the plane. It went almost totally unnoticed until a French translation was published a hundred years later. Meanwhile the idea was attributed to Jean-Robert Argand, who wrote it up independently in 1806. Since that time the geometric interpretation of complex numbers has commonly become known as the Argand diagram. Another pioneer of the theory of complex numbers was Carl Friedrich Gauss. In his doctoral dissertation of 1799 he addressed a problem that had concerned mathematicians since the early eighteenth century. Initially it had been widely believed that, just as the solutions of real quadratic equations could lead to new ‘complex’ numbers, so would solutions of equations with complex coefﬁcients lead to even more kinds of new numbers. But Jean d’Alembert (1717–83) conjectured that complex numbers alone sufﬁce. Gauss conﬁrmed this in the ‘fundamental theorem of algebra’ – every polynomial equation has a complex root. At ﬁrst he proved it in the purely real form that any real polynomial factorises into linear and quadratic factors, avoiding explicit use of imaginaries; later he treated the general case. By 1811 he viewed the complex numbers as points in the plane, saying so in a letter to Friedrich Bessel. In 1831 he published full

0.2 The Origins of Complex Analysis

5

details of his representation of complex numbers, which had begun to acquire an air of respectability. In 1837, nearly three centuries after Cardano’s use of ‘imaginary numbers’, William Rowan Hamilton published the deﬁnition of complex numbers as ordered pairs of real numbers subject to certain explicit rules of manipulation. (In the same year Gauss wrote to Wolfgang Bolyai that he had developed the same idea in 1831.) At last this placed the complex numbers on a ﬁrm algebraic basis.

0.2

The Origins of Complex Analysis Unlike the gradual emergence of the complex number concept, the development of complex analysis seems to have been the direct result of the mathematician’s urge to generalise. It was sought deliberately, by analogy with real analysis. However, the mathematicians of the period tended to assume that everything in real analysis must automatically be meaningful in the complex case, so the main question must be how ‘the’ complex version behaves. That there might not be a complex version, or several alternatives, was seldom appreciated, as the controversy over log(−x) illustrates. As noted above, there are early traces of analytic operations on complex functions in the work of Bernoulli, Leibniz, Euler, and their contemporaries. In his 1811 letter to Bessel, Gauss shows that he knew the basic theorem on complex integration around which complex analysis was subsequently built. In real analysis, when we integrate a function f between limits a and b, to get

²

a

b

f (x)dx

the limits fully specify the integral. But in the complex case, where a and b represent points in the plane, it is also necessary to specify a deﬁnite path from a to b, and to ‘integrate along the path’. The question is: to what extent does the value of the integral depend on the chosen path? Gauss says: ³ I afﬁrm now that the integral f (x)dx has only one value even if taken over different paths, provided f (x) . . . does not become inﬁnite in the space enclosed by the two paths. This is a very beautiful theorem whose proof . . . I shall give on a convenient occasion.

It seems the occasion never arose. The crucial step of publishing a proof of this result was taken in 1825 by the man who was to occupy centre stage during the ﬁrst ﬂowering of complex analysis: Augustin-Louis Cauchy. After him, this result is called ‘Cauchy’s Theorem’. In Cauchy’s hands the basic ideas of complex analysis rapidly emerged. For a complex function to be differentiable, it must have a very specialised nature: its real and imaginary parts must satisfy certain properties called the Cauchy–Riemann Equations. He showed that contour integrals of differentiable functions have the property noted privately by Gauss. Further, if an integral is computed along a path that winds round points where the function becomes inﬁnite, Cauchy showed how to compute this integral using the ‘theory of residues’. The latter requires no more than the calculation of a

6

The Origins of Complex Analysis, and Its Challenge to Intuition

constant, called the ‘residue’ of the function, at each exceptional point, and knowing how many times the paths winds around that point. The precise route of the path does not matter at all – only how it winds round these exceptional points. Power series turned out to be important in the theory, and other workers extended these ideas. Pierre-Alphonse Laurent introduced ‘Laurent series’ involving negative powers in 1843. In this formulation, near an exceptional point z0, a differentiable function is expressed as a sum of two series

= [a0 + a1 (z − z0 ) + · · · + an(z − z0)n + · · · ] +[b1 (z − z0 )−1 + · · · + an (z − z0 )−n + · · · ] The residue of f (z) at z = z0 is then just the coefﬁcient b1 . Using the theory of residues, f (x)

the computation of complex integrals often proved to be far simpler than could ever have been dreamed. Cauchy’s deﬁnition of analytic ideas such as continuity, limits, derivatives, and so on, were not the same as those we use today. He based them on inﬁnitesimal notions, which fell into disrepute in the late nineteenth century – though recent developments in ‘nonstandard analysis’, and a new theory we present in Chapter 15, show that we may have been over-hasty in judging Cauchy’s ideas. Moreover, Cauchy’s concept of ‘inﬁnitesimal’ was a variable quantity that approaches zero as closely as we please, not a ﬁxed quantity. See Tall and Katz [24] for detailed discussion and educational implications. A rigorous treatment was devised by Karl Weierstrass (1815–97) using deﬁnitions which are still regarded as fundamental, the ‘epsilon-delta’ formulation. Weierstrass founded his whole approach on power series. However, the geometric viewpoint was sorely lacking in his work (at least as published). This deﬁciency was remedied by farreaching ideas introduced by Bernhard Riemann (1826–66). In particular, the concept of a ‘Riemann surface’, which dates from 1851, treats many-valued functions by splitting the complex plane into multiple layers, on each of which the function is single-valued. The crucial feature is how the layers join up topologically. From the mid-nineteenth century onwards, the progress of complex analysis has been strong and steady, with many far-reaching developments. The fundamental ideas of Cauchy remain, now reﬁned and clothed in more recent topological language. The abstruse invention of complex numbers, once described by our mathematical forebears as ‘impossible’ and ‘useless’, has become part of an aesthetically satisfying theory with eminently practical applications in aerodynamics, ﬂuid mechanics, electronics, control theory, and many other areas. Since the ﬁrst edition of this book, formal theory has also evolved so that Cauchy’s ideas of inﬁnitesimals can be visualised as points on an extended number line, which we describe in our new Chapter 15.

0.3

The Puzzle We return to our historical puzzle. Why was the development of complex numbers so laboured and hesitant, whereas that of complex analysis was explosive? We suggest

0.4 Is Mathematics Discovered or Invented?

7

a possible answer (only personal opinion and thus open to dispute). It is somewhat different from the ‘foundations + breakthrough’ explanation offered earlier. Looking at the early history of complex numbers, the overall impression is of countless generations of mathematicians beating out their brains against a brick wall in search of – what? A triviality. The deﬁnition of complex numbers as ordered pairs of points (x, y), or as points in the plane, was obtained over and over and over again. It is even implicit in Bombelli’s work; it is there for all to see in Wallis’s; it crops up again by way of Wessel, Argand, and Gauss. Morris Kline remarks on page 629 of [11]: That many men – Cotes, de Moivre, Euler, and Vandermonde – really thought of complex numbers as points in the plane follows from the fact that all, in attempting to solve xn − 1 = 0, thought of solutions . . . as the vertices of a regular polygon.

If the problem has such a simple solution, why was this not recognised sooner? Perhaps the early mathematicians were not so much seeking a construction for complex numbers as a meaning, in the philosophical sense: ‘what are complex numbers?’ However, the development of complex analysis showed that the complex number concept was so useful that no mathematician in his right mind could possibly ignore it. The unspoken question became ‘what can we do with complex numbers?’, and once that had been given a satisfactory answer, the original philosophical question evaporated. There was no jubilation at Hamilton’s incisive answer to the 300-year old foundational problem – it was ‘old hat’. Once mathematicians had woven the notion of complex numbers into a powerful coherent theory, the fears that they had concerning the existence of complex numbers became unimportant, because mathematicians lost interest in that issue. There are other cases of this nature in the history of mathematics, but perhaps none is more clear-cut. As time passes, the cultural world-view changes. What one generation sees as a problem or a solution is not interpreted in the same way by a later generation. It is worth bearing this in mind when thinking about the historical development of mathematics. To interpret history solely from the viewpoint of the current generation may easily lead to distortion and misinterpretation. What this explanation omits is any discussion of why mathematicians lost interest in the meaning of complex numbers. And that leads to a question that sheds a different light on the historical development, which we now discuss.

0.4

Is Mathematics Discovered or Invented? Students trying to understand new concepts are in a similar position to the pioneers who ﬁrst investigated them. At any stage in our education, we build not just on our current knowledge, but on a variety of beliefs and intuitions that are often vague, and may not be consciously recognised. As a trivial example, children familiar with counting numbers may ﬁnd it hard to adapt their thinking to negative numbers, or rational numbers. When faced with questions like ‘what is 3 minus 7?’ or ‘what is 3 divided by 7’, intuition based solely on whole numbers leads to the answer ‘can’t be done’. That makes it hard

8

The Origins of Complex Analysis, and Its Challenge to Intuition

to understand −4 or 3/7. In fact, these is not really trivial examples, because the world’s top mathematicians, centuries ago, were just as confused by the question ‘what is the square root of minus one?’ Even their terminology – ‘imaginary’ – reveals how puzzled they were. Intuitively they considered numbers to be ‘real’ – not in the sense we now use to distinguish real from complex, but as direct representations of real measurements. The new objects behaved like numbers in many ways, but they seemed not to correspond directly to reality. In such circumstances, it can be tempting to discard existing intuition completely. But it is more sensible to adapt the intuition to ﬁt the new circumstances. It is much easier to do arithmetic with negative numbers or fractions if you remember how to do it with whole numbers; it is much easier to do algebra with complex numbers if you bear in mind how to do it with real numbers. So the trick is to sort out which aspects of existing intuition remain valid, and which need to be reﬁned into a broader kind of understanding. One way to approach this issue is to take seriously a question that is often asked but seldom answered satisfactorily: is mathematics discovered or invented? One answer is to dismiss the question, and agree that neither word is entirely appropriate; moreover, they are not mutually exclusive. Most discoveries have elements of invention, most inventions have elements of discovery. Galileo would not have discovered the moons of Jupiter without the invention of the telescope. The telescope could not have been invented without discovering that sand could be melted to make glass. But leaving such quibbles aside, we can make a rough distinction between discovery, which is ﬁnding something that is already there but has not hitherto been noticed, and invention, which is a creative act that brings into being something that has not previously existed. There is a case to be made that in this sense, mathematicians invent new concepts but then discover their properties. For example, complex integration is all about ‘paths’ in the complex plane. Intuitively, a path is a line drawn by moving the hand so that the pencil remains in contact with the paper – no jumps. We might choose to formalise this notion as a continuous curve – the image of a continuous map from a real interval to the complex plane. We might be interested in how the pencil point moves along this curve, which requires the map itself, not just its image. Sometimes we might wish the path to be smooth – to have a well-deﬁned tangent. As it happens, we need all of these notions. Intuitively, they are all based on the same mental image. Formally, they are all very different. They have different deﬁnitions, different meanings, and different properties. A smooth path always has a meaningful length, for instance; a continuous path may not. The deﬁnitions we settle on in this book ﬁt conveniently into the standard ideas of analysis, but they are not built into the fabric of the universe. We chose them, and by so doing we invent concepts such as ‘path’, ‘curve’, and ‘smooth’. On the other hand, once a concept has been invented, we cannot invent its properties. When we also invent the concept ‘length’, we discover that every smooth path has ﬁnite length. We cannot ‘invent’ a theorem that the length of a smooth path can be inﬁnite. If we weaken ‘smooth’ to ‘continuous’, however, we can discover that inﬁnite lengths are possible; indeed, ‘length’ need not have a sensible meaning at all. In short: invention

0.4 Is Mathematics Discovered or Invented?

9

opens up new mathematical territory, but exploring it leads to discoveries. We may not know what things are present in the territory, but we do not get to choose them. Sometimes – in fact, very often – we discover that our inventions have features that we neither expected nor intended them to have. We discover, perhaps to our dismay, that the image of a smooth path can have a right-angled corner, see Section 6.7. We did not expect that: a corner does not feel ‘smooth’. But its possibility is a direct consequence of the deﬁnition we invented. When this kind of thing happens, we have two choices. Accept the surprises as the price for having a nice, tidy deﬁnition; or rule them out by changing the deﬁnition – inventing a more comfortable alternative. In practice we often do both, by giving the alternative a different name. Here we could (and do) deﬁne a ‘regular path’ to be a smooth path γ : [a, b] → C for which γ ² (t) ³ = 0 whenever t ∈ [a, b]. Now the image cannot have a sharp corner. On the other hand, every theorem about regular paths must now take account of the consequences of that extra condition. We also have to remember that some theorems may be valid for regular paths but not for smooth paths, and so on. As we move from intuitive ideas to formal ones, we also reﬁne our intuition so that it matches the formal theory better. Formal calculations start to make sense, not just as strings of symbols that follow from previous strings, but as meaningful statements that agree with our new intuitions. From this point of view, the history of complex analysis is the story of intuition co-evolving with an increasingly formal approach. This suggests that mathematicians lost interest in the meaning of complex numbers when they incorporated them into their intuitive assumptions and beliefs. With the apparent conﬂicts resolved by these reﬁned intuitions, they were free to push the subject forward, no longer worried that it did not make logical sense. When a mathematical area ‘settles down’ into a mature theory, there is a broad consensus that certain concepts provide the most convenient route through the material. These concepts then become standard – things like ‘continuous’, ‘connected’, and so on. They get taught in lecture courses and printed in books. We may start to feel that the standard deﬁnitions are the only reasonable ones. Even so, we are always free to work with different concepts if that seems sensible, or even to modify deﬁnitions while retaining the same name – though that can be dangerous. Today’s concept of continuity is quite different from what it was in the time of Euler, but we use the same word; we just bear in mind that it now has a speciﬁc technical meaning. A historian reading Euler would need to be on their guard. It is also worth remarking that many mathematical concepts seem more natural to us than others. Counting numbers are very natural (we even call them the ‘natural numbers’). The number i was bafﬂing for centuries (and was called ‘imaginary’ as a result). Our culture, our society, and even our senses, predispose us towards certain concepts. Euclid’s points and lines correspond to early stages of the processing of images sent from the retina to the visual cortex. Newton’s concept of acceleration being related to an applied force reﬂects the way our ears sense accelerations and make us ‘feel’ a push – a force. It then becomes easy to imagine that mathematics somehow already exists in a realm outside the natural world. Even if humans invented numbers, in retrospect they seem

10

The Origins of Complex Analysis, and Its Challenge to Intuition

such a natural idea that surely they were just hanging around waiting to be invented. If so, that is more like discovery. This view is often called Platonism: the idea that mathematical concepts already exist in some ideal form in some kind of world outside the physical universe, and mathematicians merely discover how these ideal forms work. The contrary view is that mathematics is a shared human construct, but that construct is by no means arbitrary, because every new invention is made in the context of existing knowledge, and every new discovery must be logically valid. A major theme of this book is that many apparently puzzling aspects of complex analysis can be made more intuitive by paying attention to the geometry of the complex plane (in a broad sense, including its topology). This brings one of the human brain’s most powerful abilities, visual intuition, into play. For this reason, we draw a lot of pictures. However, a picture, and our visual intuition, can be misleading unless we examine the unstated assumptions that they involve. By doing so, we can reﬁne out intuition and make it more reliable. For this reason, we do not just introduce important deﬁnitions and then deduce theorems that refer to them. We try to relate those deﬁnitions to intuition, to make the proofs easier to understand. Then we exhibit some of the positive results that arise, to convince you that the new concept is worth considering. And then . . . we show you that sometimes the formally deﬁned concept does not behave the way intuition might suggest. Sometimes it turns out to be useful to strengthen the deﬁnition so that it matches intuition more closely. Sometimes we reﬁne our intuition so that it matches the formal deﬁnition. Sometimes we can even do both, in which case we have to make some careful but useful distinctions. The historical events sketched earlier in this chapter offer many examples of this process. The square root of minus one went from being a puzzling idea that seemed to have no meaning to one of the most important concepts in the whole of mathematics. Along the way, mathematicians’ intuition for ‘number’ underwent a revolution. We can now to some extent short-circuit the historical debates – what were hang-ups then need not be hang-ups now – but when a new idea puzzles us, and doesn’t seem to make sense until we ﬁnally sort it out, it is helpful to remember that the mathematical pioneers often experienced exactly the same feelings, for much the same reasons.

0.5

Overview of the Book It is often useful to set the development of a mathematical theory in its historical context, but it is not always necessary to ﬁght the historical battles again. In this text we give honour where we can to those pioneers who carved their way through uncharted mathematical territory. But more recent developments let us see the theory itself in a new light. To the modern ear the very name ‘complex analysis’ carries misleading overtones: it suggests complexity in the sense of complication. The older meaning, ‘composite’, was perhaps appropriate when the ‘real part’ of a complex number had a quite different status from that of the ‘imaginary part’. But nowadays a complex number is a perfectly integrated whole. To think of complex analysis as if it were, so to speak, two copies of real analysis, is to place undue emphasis on the algebra at the expense of the geometry,

0.5 Overview of the Book

11

which in the long run has been far more inﬂuential. And in fact complex numbers are not more complicated than reals: in some ways, they are simpler. For instance, polynomials always have roots. Likewise, complex analysis is often simpler than real analysis: for example, every differentiable function is differentiable as often as we please, and has a power series expansion. In preparing our approach to the subject we have adopted two basic organising principles. The ﬁrst is the direct generalisation of real analysis to the complex case. Deﬁnitions, of limits, continuity, differentiation, and integration are natural extensions of the corresponding real notions. Since nowadays any student taking a course in complex analysis may be assumed to have made a study of the real counterpart, many battles have already been won. We can refer students to their accumulated knowledge, pausing only to phrase it appropriately. This saves time and energy, allowing us to proceed straight to the heart of the subject, where the interesting differences occur. Invariably this happens because the plane has a richer geometry than the line, and this leads to our second major organising principle: geometric insight is valuable and should be cultivated. Of course this insight must be translated into sound formal arguments; this can often be done using modern topological notions. From these two principles, a straightforward approach to complex analysis emerges. First, complex numbers are deﬁned formally as ordered pairs of real numbers, giving them a geometric interpretation as points in the plane. The topology of complex numbers is then a natural consequence of plane topology. In quick succession it is possible to derive complex generalisations of the notions of continuity, limits, and differentiation, with particular emphasis on power series, which play a central role later. A study of the complex exponential function, deﬁned by the usual power series, reveals the intimate connection between this function and the trigonometric functions (also considered as power series). After generalising the notion of integration, the logarithm can be viewed either as the inverse function of the exponential, or as the integral log z

=

²

dz z

suitably interpreted. Either approach has to deal with the multivalued nature of the complex logarithm. This arises because the complex exponential has period 2π i, so cannot be one-one. Resolving these issues involves close links between geometric intuition and formal analysis. At this stage Cauchy’s Theorem is presented in various guises, and the use of integration leads to a proof that every differentiable function can be expressed as a power series. More generally, Laurent series (using positive and negative powers) take care of isolated points where functions become inﬁnite, and lead to the powerful ‘theory of residues’ for calculating complex integrals, summing series, and counting zeros of equations in a given region of the complex plane. Returning to geometric ideas, complex analysis has many practical applications. Today it is widely used by physicists and engineers, in many different contexts. In particular, it has proved invaluable in two-dimensional potential theory. The geometric ideas

12

The Origins of Complex Analysis, and Its Challenge to Intuition

of Riemann can be viewed in terms of modern topology, to give a global insight into ‘many-valued’ functions (such as the logarithm) and open up new areas of progress. In this second edition of the book, we continue by presenting a formal set-theoretic approach to inﬁnitesimals that has evolved since the ﬁrst edition was published 35 years ago. It offers a new vision of complex analysis that includes both the analytic epsilondelta approach of Riemann and the inﬁnitesimal ideas of Cauchy in a broader overall theory. Next, we revisit Cauchy’s Theorem in the context of homology theory. Homology is a topological property of the domain of the function, and it detected the presence of holes. These holes are obstacles that cause integrals of complex functions to depend on the chosen path. Using step paths, we reformulate complex integration over ‘cycles’ in a domain. These are formal integer combinations of closed loops, so they form an abelian group. The subgroup of ‘boundaries’ has the property that the integral of any continuous function over a boundary is zero. So the difference between cycles and boundaries controls how integrals depend on the choice of path. The corresponding algebraic object is the quotient group of the group of cycles modulo the subgroup of boundaries, and this is the (ﬁrst) homology group of the domain. It provides a formal algebraic interpretation of how integrals depend on the choice of path. This chapter provides a gentle introduction to homology in its simplest (old-fashioned) form, though even this approach requires some mathematical sophistication. The topological ideas shed light on the general area surrounding Cauchy’s Theorem. Finally, to show that complex analysis is still alive and kicking in the modern era, Chapter 17 provides a simpliﬁed overview of a few more recent developments. These include the still-unsolved Riemann Hypothesis, modular functions, generalising complex analysis to several variables (where strange new phenomena occur), to complex manifolds (multidimensional ‘surfaces’ with a complex structure, generalising Riemann surfaces), and the iteration of complex maps, or complex dynamics, which leads to remarkable fractal structures such as Julia sets and the Mandelbrot set.

1

Algebra of the Complex Plane

‘The Divine Spirit found a sublime outlet in that wonder of analysis, that portent of the ideal world, that amphibian between being and not-being, which we call the imaginary root of negative unity.’ So said Leibniz in 1702 – though he may have let his eloquence √ run away with him. The current view of −1 is a little more prosaic, though the uses made of it are at least as inspiring. The logical status of complex numbers, which caused so much distress during the eighteenth century, is now seen to be very much on a par with that of the ‘real’ numbers. What puzzled the ancients was the obvious artiﬁciality and abstraction of the complex number system, in contrast to the apparently natural and concrete real number system. But the mathematician of today sees even real numbers as possessing a similar artiﬁciality and abstraction. In this chapter we discuss the construction of a system of numbers that contains the familiar real numbers and permits the solution of the equation x2 = −1. This system is known as the complex numbers. Many readers will already know the contents of this chapter: they should read it through rapidly to check such items as notation, and pass on at once to the next. There is a natural geometric representation of complex numbers as a plane, analogous to that of the reals as a line. The extra freedom inherent in the plane gives the whole subject a very geometric ﬂavour, which it is our intention to keep to the fore in the development of the theory.

1.1

Construction of the Complex Numbers We begin with the deﬁnition that emerged from the insights of Wallis, Wessel, Argand, Gauss, and Hamilton: D E FI N I T I O N 1.1.

A complex number is an ordered pair (x, y) of real numbers. Addition and multiplication of complex numbers are deﬁned by: (x1 , y1) + (x2, y2 ) = (x1 + x2 , y1 + y2)

(x1, y1 )(x2, y2 ) = (x1 x2 − y1 y2, x1 y2 + x2y1 )

For example, (3, 5)(2, 7)

= (3 · 2 − 5 · 7, 3 · 7 + 5 · 2) = (−29, 31)

(1.1) (1.2)

14

Algebra of the Complex Plane

This deﬁnition is the culmination of several centuries of struggle to understand complex numbers, and it shows how elusive a simple idea can be. Before we see what these pairs √ have to do with −1, however, let us establish some of their properties. 1.2. The set of complex numbers, with the operations deﬁned by (1.1, 1.2), is a ﬁeld. That is, the following axioms hold: if z1 = (x1, y1 ), z2 = (x2, y2 ), and z3 = (x3, y3) are complex numbers, then

THEOREM

(a) Addition and multiplication are commutative: z1 + z2 z1 z2

= z2 + z1 = z2 z1

(1.3)

(b) Addition and multiplication are associative: (z1 + z2) + z3 (z1z2 )z3

= z1 + (z2 + z3 ) = z1 (z2z3 )

(1.4)

(c) There is an additive identity (0, 0): z1 + (0, 0) = z1

(1.5)

(d) There is a multiplicative identity (1, 0): z1 (1, 0) = z1

(1.6)

(e) Each element has an additive inverse: (x, y) + (−x, −y) = (0, 0)

(1.7)

(f) Each element other than (0, 0) has a multiplicative inverse:

±

(x, y)

² − y , = (1, 0) x2 + y2 x2 + y2 x

(1.8)

(g) Multiplication distributes over addition: z1(z2 + z3) = z1 z2 + z1 z3

(1.9)

Proof. All assertions (a)–(g) are direct consequences of (1.1) and (1.2), using only the ﬁeld properties of the set R of real numbers. For example, (1.9) holds because z1(z2 + z3) = (x1 , y1 )(x2 + x3 , y2 + y3)

= (x1 (x2 + x3) − y1(y2 + y3 ), x1(y2 + y3 ) + y1 (x2 + x3 )) = (x1 x2 + x1x3 − y1y2 − y1 y3 , x1 y2 + x1 y3 + y1 x2 + y1x3 )

and z1 z2 + z1 z3

= (x1, y1 )(x2, y2 ) + (x1 , y1 )(x3 , y3 ) = (x1x2 − y1y2 , x1y2 + y1x2 ) + (x1 x3 − y1y3, x1 y3 + y1x3 ) = (x1x2 − y1y2 + x1 x3 − y1 y3, x1 y2 + y1 x2 + x1y3 + y1x3 )

which, by real algebra, is the same ordered pair. The reader should supply similar proofs for the remaining assertions. The symbol C is used for the ﬁeld of complex numbers.

1.2 The x

1.2

The x

+ iy Notation

15

+ iy Notation

The symbol commonly used for a complex√ number is not (x, y) but x + iy. This notation goes back to Euler, who used i to denote −1 in 1777, though the notation was ﬁrst used consistently by Gauss. To recover this notation, we proceed as follows. First note that since (x1 , 0) + (x2 , 0) = (x1 + x2, 0) (x1, 0)(x2 , 0) = (x1 x2 , 0)

we may identify a complex number (x1 , 0) with the real number x1 . More pedantically the map (x1, 0) ± → x1 deﬁnes an isomorphism between the set of complex numbers of the form (x1, 0) and the ﬁeld R of real numbers. Now deﬁne i = (0, 1)

Then x + iy = (x, 0) + (0, 1)(y, 0)

= (x, y)

by (1.1) and (1.2)

Finally, observe that

= (0, 1)(0, 1) = (0 · 0 − 1 · 1, 0 · 1 + 1 · 0) = (−1, 0) = −1 √ In this sense, we may say that i = −1. The x + iy notation is more convenient, and will be used from now on. (Sometimes we use x + yi instead. By (1.3) this represents the same number. In particular, we use this form when x, y are speciﬁc real numbers, because 1 + 2i looks more sensible than 1 + i2.) i2

Algebraic computations in this notation are easy. They use all the normal algebraic rules, plus the rule i2 = −1. So to multiply, we work out (x1 + iy1 )(x2 + iy2 ) = x1x2 + x1 iy2 + iy1x2 + iy1 iy2

= x1x2 + i(x1 y2 + y1 x2) + i2 y1y2

But i2

= −1, so this becomes

x1 x2 − y1y2 + i(x1 y2 + y1 x2)

This computation, of course, explains the choice of the multiplication formula (1.2). The addition formula (1.1) comes the same way but is easier. The deﬁnition by (1.1) and (1.2) is thus a very sneaky piece of hindsight. The formula (1.8) for inverses may also be derived as follows: 1 x − iy x − iy = = x + iy x + iy x − iy x2 + y2 1

16

Algebra of the Complex Plane

Example 1.3. Express

We have

1.3

2 + 3i 1 + 2i

2 + 3i in the form x + iy. 1 + 2i

= 12 ++ 2i3i 11 −− 2i2i = 2 + 6 + i(5 −4 + 3) = − 58 − 5i

A Geometric Interpretation Since ordered pairs (x, y) provide coordinates in the plane R2, we can visualise C as a plane, with the number x + iy corresponding to the point (x, y) as in Figure 1.1. The identiﬁcation of (x, 0) with x ∈ R then amounts to considering the real numbers as forming the real axis in the plane, as in Figure 1.2. The y-axis, at right angles to this, is the imaginary axis. This geometric representation is often called the Argand diagram or the Gauss plane. Since so many other mathematicians (especially Wessel) have justiﬁable claims to it, we avoid the danger of giving undue credit to any of them by referring to it as the complex plane. In purely geometric terms, of course, it is just the real plane R2 , but interpreted as C it has the additional algebraic structure of a ﬁeld, not just a vector space over R. It is this extra structure that gives the complex plane its special qualities.

Figure 1.1 Visualising a complex number as a point in the plane

Figure 1.2 Identifying x

+ 0i ∈ C with x ∈ R.

R2 .

1.5 The Modulus

1.4

17

Real and Imaginary Parts Given a complex number z part, using the notation

= x + iy, we call x the real part of z and y the imaginary x = re(z)

y = im(z)

Both are real numbers: the coordinates of z in the complex plane.

1.5

The Modulus The modulus, or absolute value, of a real number x is deﬁned to be

³

|x| = −xx

if x ≥ 0 if x < 0

As it stands, there is no obvious generalisation to complex numbers, because (see Section 1.8 below) there is no useful ordering on C. However, we can interpret | x| geometrically as the distance from x to the origin of the real number line. This translates directly to the complex plane, leading to the deﬁnition

| z| =

´

x2 + y2

for the modulus, or absolute value, of a complex number z = x + iy. Here we mean the positive square root: since x2 + y2 is always a positive real number, the formula deﬁnes |z| as a real number. THEOREM

1.4. The modulus has the following properties:

| z1 + z2 | ≤ | z1 | + | z2 | |z1z2 | = |z1||z2| ||z1 | − |z2 || ≤ |z1 − z2|

(1.10) (1.11) (1.12)

Proof. Property (1.11) follows at once from the deﬁnitions. The triangle inequality (1.10) is a little harder to prove directly, although its geometric interpretation (Figure 1.3) is the obvious fact that one side of a triangle is no longer than the sum of the lengths of the other two sides. To prove (1.10) note that, since both sides are positive, it is equivalent to (|z1 + z2 |)2

≤ (|z1| + |z2|)2

which takes the form (x1 + x2 )2 + (y1 + y2)2 where z1

≤ |z1 |2 + 2|z1 ||z2 | + |z2 |2

= x1 + iy1, z2 = x2 + iy2 . Simplifying, this holds if and only if x1x2 + y1y2 ≤ |z1||z2 |

18

Algebra of the Complex Plane

Figure 1.3 Geometry for the triangle inequality.

Since the right-hand side is positive, we may square again, and the desired inequality follows from (x1x2 + y1 y2 )2

≤ |z1|2|z2 |2

But

|z1|2|z2 |2 − (x1 x2 + y1y2)2 = (x21 + y21 )(x22 + y22) − (x1x2 + y1y2 )2 = (x1 y2 − x2 y1)2 which is positive. Property (1.12) is a consequence of (1.10). This implies that

|z1 − z2| + |z2 | ≥ |z1| so

| z1 − z2 | ≥ | z1 | − | z2 |

Swapping z1 and z2 we also have

|z1 − z2 | = |z2 − z1| ≥ |z2 | − |z1| Combining the two inequalities yields

||z1| − |z2 || ≤ |z1 − z2| 1.6

The Complex Conjugate If z = x + iy, its complex conjugate is z¯

= x − iy

Geometrically, this is obtained by reﬂecting z in the x-axis, Figure 1.4.

1.7 Polar Coordinates

19

Figure 1.4 Geometry for the complex conjugate.

The following properties are easy to verify directly: z1 + z2

= z¯1 + z¯2 z1z2 = z¯1z¯2 re(z) = 21 (z + z¯) 1 (z − z¯) im(z) = 2i 2 |z| = zz¯ z¯ ∈ R if and only if z = z¯

(1.13) (1.14) (1.15) (1.16) (1.17) (1.18)

Properties (1.13, 1.14) have the important implication that the complex conjugate of any polynomial expression in complex numbers z1 , z2 , . . . , zn can be obtained by writing a bar over each individual coefﬁcient or variable in the expression. This is easily proved by induction. For example,

= 5¯ z¯1z¯2 − z¯3 7 + 2¯ ¯iz¯1 = 5z¯1z¯2 − z¯3 7 − 2iz¯1 since 5, 2 are real and i is imaginary, so 5¯ = 5, 2¯ = 2, ¯i = −i. 5z1 z2 − z37 + 2iz1

1.7

Polar Coordinates The expression x+iy for a complex number is intimately related to Cartesian coordinates (x, y) in the plane. It turns out often to be useful to work with polar coordinates (r, θ ), which we recall correspond to a point distance r from the origin making an angle θ measured from the positive x-axis in an anticlockwise direction, Figure 1.5. Of course we measure θ in radians. These coordinate systems are related as follows: x y

= =

r cos θ r sin θ

(1.19)

20

Algebra of the Complex Plane

Figure 1.5 Polar coordinates.

Therefore r

=

´

x2 + y2

= |z|

where z = x + iy. Finding θ is slightly trickier because it is not unique. Any value of θ for which (1.19) holds is called an argument of z. The article ‘an’ is used to reﬂect the lack of uniqueness: if θ is an argument then so is θ + 2kπ for any integer k. With the understanding that θ is unique only up to multiples of 2π , we may use the notation

θ = arg z

Often the choice of θ is rendered unique by imposing some convention: for example, we may insist that θ is chosen in the interval [0, 2 π ), or in ( −π , π ]. The unique value of θ in the interval (−π , π ] is known as the principal value of the argument. (We follow standard practice in taking this particular interval. Its main advantage is that θ then behaves nicely near the positive real axis, where θ = 0. But this is a technical point that only acquires importance much later. The non-uniqueness of θ is a phenomenon with tremendous ramiﬁcations in the theory, as we shall see.) With r, θ deﬁned as above, z = x + iy = r(cos θ + i sin θ )

The expression cos θ + i sin θ is of considerable importance in complex analysis. In Chapter 5 we relate it to the complex exponential function.

1.8

The Complex Numbers Cannot be Ordered The real numbers may be given an ordering (the usual one, properties the following: If x ² = 0 then either x > 0 or

>) which has among its

− x > 0, but not both If x, y > 0 then x + y > 0, xy > 0

(1.20) (1.21)

1.9 Exercises

21

No such ordering can be deﬁned on the complex numbers. Suppose for a contradiction that one can. Since i ² = 0, (1.20) implies that either i > 0 or −i > 0. Then (1.21) implies that either −1 = i · i > 0 or −1 = (−i) · (−i) > 0. At the same time, 1 = (−1)2 > 0. But then both 1 and −1 are greater than 0, contrary to (1.20). It is therefore not possible to use inequalities, analogous to those for reals, when discussing complex numbers. Any inequality that occurs must involve only real numbers, possibly related to the given complex numbers. For example, if z ∈ C then z>1

makes no sense, but either of or

|z| > 1 re(z) > 1

is acceptable. (They do not mean the same thing!) As a convention, if we write a statement such as

ε>0

this will automatically imply that ε is assumed to be a real number.

1.9

Exercises 1. Check in full detail that the complex numbers C form a ﬁeld under the operations of addition and multiplication deﬁned in (1.1, 1.2). 2. In Figure 1.6 the black dots represent three complex numbers u, v, w (as marked). The circle is the unit circle |z| = 1. The open dots a, b, c, d, e, f , g, h represent (in some order) the numbers u + v, u + w, v + w, u + v + w, uv, uw, vw, uvw. Which is which?

Figure 1.6 Data for Exercise 2.

22

Algebra of the Complex Plane

3. By writing z in the form z = a + bi, ﬁnd all solutions z of the following equations: (i) (ii) (iii) (iv) (v)

z2 = −5 + 12i z2 = 2 + i (7 + 24i)z = 375 z2 − (3 + i)z + (2 + 2i) = 0 z2 − 3z + 1 + i = 0

4. If λ is a positive real number, show that

{z ∈ C : |z| = λ|z − 1|}

is a circle, unless λ takes one particular value (which?) 5. Draw the set of points

{z ∈ C : re(z + 1) = |z − 1|}

by substituting z = x + iy and computing the real equation relating x and y. Now note that re(z + 1) is the distance from z to the line y = −1, and | z − 1| is the distance between z and 1. Compare with the classical ‘focus-directrix’ deﬁnition of a parabola: the locus of a point equidistant from a ﬁxed line (here y = − 1) and a ﬁxed point (here (x, y) = (1, 0)).

6. Draw the set of all z ∈ C satisfying the following conditions: (i) (ii) (iii) (iv) (v) (vi) (vii) (viii)

re(z) > 2 1 < im(z) < 2 1 < im(z − i) < 2 |z| < 2 |z| > 1 1 < |z| < 2 |z − 1| < 1 |z − 1| < |z + 1|

7. Draw the set of all z ∈ C satisfying the following conditions: (i) (ii) (iii) (iv)

z¯z = 1 z + i¯z + 1 + i = 0 z + ¯z + 2 = 0 z + ¯z + 2i = 0

8. Let r, s, θ , φ be real. Let

z = r(cos θ + i sin θ )

w = s(cos φ + i sin φ)

Form the product zw and use the standard formulas for cos(θ + φ), sin( θ + φ ) to show that arg(zw) = arg(z) + arg(w) (for any values of arg on the right, and some value of arg on the left). By induction on n, derive De Moivre’s Theorem (cos θ for all natural numbers n.

+ i sin θ )n = cos nθ + i sin nθ

1.9 Exercises

23

Figure 1.7 Data for Exercise 10.

Specialise to the case n = 3 and recover the usual formulas for cos 3 θ and sin 3θ in terms of cos θ and sin θ .

9. Use De Moivre’s Theorem (Exercise 8) and the substitution z = r(cos θ + i sin θ ) to show that the equation z3 = 1 has three √ distinct √ complex roots. Find them. Compute the square roots of 1 + i 3, 3 − i, and 1 + i, and the cube roots of √ 3 + i, 1 − i, i. Sketch these points in the complex plane. 10. In earlier textbooks, multiplication of complex numbers is often deﬁned as follows. Given two complex numbers z1 , z2 , represent them by points A and B in the complex plane; and let O, U be the points z = 0, 1 respectively, Figure 1.7. Draw triangle OBC similar to triangle OUA (where ∠BOC = ∠UOA, ∠OBC = ∠OUA). Then z1z2 is represented by the point C so constructed. Using the fact that |z1z2 | = |z1||z2| , and the result of Exercise 5, show that this construction agrees with our deﬁnition (1.2).

11. Deﬁne a square root z of a complex number z to be any complex number w such that w2 = z. Prove that every non-zero complex number has exactly two square roots, and give formulas for them in terms of re z and im z. If a, b, c ∈ C with a ² = 0, show that the solutions of the quadratic equation az 2 + bz + c = 0

are precisely

−b ³ z=

b2 − 4ac 2a 12. Use De Moivre’s Theorem (Exercise 8) to compute cos 5θ and sin 5θ in terms of cos θ and sin θ . 13. Prove that De Moivre’s Theorem remains true if n is a negative integer.

14. Deﬁne a kth root k z to be any w such that wk √ ﬁnd an expression for k r(cos θ + i sin θ ).

= z. Use De Moivre’s Theorem to

2

Topology of the Complex Plane

In this chapter we collect together the basic topological ideas required for our study of complex analysis. The list is not very demanding. Some items are needed to handle differentiation neatly, and some are needed for integration. Differentiation is naturally set against a background of limits and continuity, and these are best dealt with on open sets. On the other hand, an interval from one complex number to another is computed along a speciﬁed path between them. A set in which any two points can be joined by a path is said to be path-connected. To be able to cope with both integration and differentiation in the simplest possible manner later on, we restrict our complex functions to be those deﬁned on open path-connected sets. Such a set is called a domain. When the set is open we often abbreviate ‘path-connected’ to ‘connected’. The term ‘connected’ is used in point-set topology with a speciﬁc technical meaning, but for open sets in C it is equivalent to being path-connected. So this abbreviation does no harm. Domains can have exotic shapes and paths can wiggle around a great deal. To be able to appeal to geometric intuition without our imagination having to work overtime thinking about such complications, we use a carefully conceived technical device called the Paving Lemma. We show in this lemma that a path in an open set (in particular, a domain) can be subdivided into a ﬁnite number of smaller pieces in such a way that each piece is contained in a disc within the open set, thus ‘paving’ the path with discs, see Figure 2.1. A disc is the interior of a circle, which is geometrically very simple: for instance, any two points in it can be joined by a straight line. Joining the end points of pieces of the original path in each disc paving it, we obtain a new path made up of straight line segments, still lying in the open set and joining the ends of the original path. We see, therefore, that given any path whatsoever between two points in an open set, no matter how much the path twists and turns, there is an alternative path in the open set, between the same points, that is made up of a ﬁnite number of straight line segments. We can even insist that the segments are parallel to the real or imaginary axis, giving a step path in the open set. To do so, take a suitable step path inside each paving disc and join them together, see Figure 2.2. With techniques such as this we can use the Paving Lemma to illuminate complex analysis, yielding fully rigorous proofs linked ﬁrmly to geometric intuition.

Topology of the Complex Plane

25

Figure 2.1 A domain (shaded) and a path in the domain (solid curve). Circles show a ﬁnite set of

discs inside the domain, whose union contains the path.

Figure 2.2 Replacing the path in Figure 2.1 by a step path.

26

Topology of the Complex Plane

2.1

Open and Closed Sets 2.1. For a complex number z0 and a positive real number ε-neighbourhood of z0 is

D E FI N I T I O N

Nε (z0 ) = {z ∈ C : |z − z0|

ε,

the

< ε}

This is an open disc of radius ε . Geometrically, Nε (z0 ) is the disc centre z0 of radius ε , Figure 2.3. A subset S ⊆ C is said to be open if for every z0 ∈ S there is a real number that Nε (z0 ) ⊆ S. We emphasise that ε may depend on z0.

ε such

| − |

δ < ε−| − |

D E FI N I T I O N

2.3. The complement of a subset S ⊆ C is

C \ S = {z ∈ C : z ∈/ S}

A subset S is closed if C \ S is open.

ε

Figure 2.3 The -neighbourhood of a point z0 .

ε

Figure 2.4 The -neighbourhood of z0 is open.

2.2 Limits of Functions

27

There is another way to characterise closed sets, using the notion of a limit point of a subset S. A complex number z0 is a limit point of S if every neighbourhood Nε (z0 ) contains a point of S not equal to z0 . In this deﬁnition, z0 does not itself have to belong to S, though it may do. The essential feature of a limit point of S is that it has points of S arbitrarily close to it. In fact, each Nε (z0 ) must contain an inﬁnite number of points of S – for if some Nε (z0 ) contains only ﬁnitely many points z1, . . . , zn of S, distinct from z0 , we can take ε1 to be the smallest of the distances |z0 − zr | . Then Nε 1 (z0) contains no points of S, a contradiction. An alternative characterisation of a closed set is: P RO P O S I T I O N

2.4. A subset S

points.

⊆ C is closed if and only if S contains all its limit

Proof. Suppose S is closed and z0 is a limit point. If z0 ∈ C \ S, which is open, then Nε (z0) ⊆ C \ S for some ε > 0, so Nε (z0) contains no point of S, contradicting z0 being a limit point. Hence z0 ∈ S. Conversely, suppose that S contains all its limit points. Then any z0 ∈ C \ S is not a limit point of S, so there exists ε > 0 such that Nε (z0 ) contains no point of S. Therefore Nε (z0) ⊆ C \ S, so C \ S is open, and S is closed. Not every point of a closed set need be a limit point of that set. For instance, if T

= {z ∈ C : z = 0 or z = 1/n for a positive integer n}

then the only limit point of T is 0. Since this is in T , the set T is closed. Points in S that are not limit points of S are said to be isolated points of S. All points of T except 0 are isolated points of T. In general, an isolated point z0 of S has a neighbourhood Nε (z0 ) that contains no points 1 of S other than z0 . For instance, in the set T , if ε = n1 − n+ then Nε (1/n) contains no 1 other points of T. On the other hand, it is clear from the deﬁnitions that every element of an open set is a limit point of that set. We will also need: The closure of a subset S ⊆ C is the intersection of all closed subsets that contain S. This is closed, and is the smallest closed subset containing S. It consists of S together with all limit points of S. D E FI N I T I O N 2.5.

2.2

Limits of Functions The notion of a limit lim f (z)

z→z 0

is analogous to the real case, and its properties follow by similar arguments. If f : S → C is an arbitrary complex function and z0 is a limit point of S, then limz →z0 f (z) = l if, given any ε > 0, there exists δ > 0 such that D E FI N I T I O N 2.6.

for all z ∈ S, 0 < | z − z0 | < δ implies | f (z) − l|

0 such that {z : |z − z0| < δ} contains no point of S. In this case condition (2.1) would be vacuously true for any l ∈ C. As in the real case, we have: P RO P O S I T I O N

2.7. If z0 is a limit point of S and limz→z 0 f (z) = l, the limit is unique.

Proof. Suppose that l ² ± = l is another candidate for the limit. Take δ1 > 0, δ2 > 0 such that

ε = 21 |l − l² | to ﬁnd

z ∈ S, 0 < | z − z0 | implies | f (z) − l | < ε z ∈ S, 0 < | z − z0 | implies | f (z) − l ²|

0 such that z ∈ S and 0 < | z − z0 | < δ implies | g(z) − k| < ε Therefore |g(z)| < | k| +ε = M, say, whenever 0 < | z − z0| . Therefore |g(z)| is bounded above near (but not necessarily at) z0 by M. Since limz →z0 f (z) = l, there exists δ1 > 0 such that z ∈ S and 0

< |z − z0| < δ1 implies | f (z) − l| < ε/(2M) Let N ∈ R such that N > |l| , so in particular N > 0. Since lim z→z g(z) = k, there exists δ2 > 0 such that z ∈ S and 0 < | z − z0 | < δ2 implies | g(z) − k| < ε/ (2N) For δ = min(δ 0, δ1 , δ 2) we then have z ∈ S and 0 < |z − z0| < δ1 implies ε M+N ε =ε | f (z) − l||g(z)| + |l||g(z) − k| < 2M 2N By (2.3), for z ∈ S and 0 < | z − z0 | < δ2, we have | f (z)g(z) − lk| < ε 0

proving (iii). To prove (iv), it is enough to prove that limz →z 0 1/g(z) = 1/k, and then to appeal to (iii) for the functions f , 1/g. For this to make sense we must show that for some δ 1 > 0, g(z) is not zero when z ∈ S and 0 < | z − z0 | < δ1 . Then 1/g(z) is deﬁned. This follows since lim z→z0 g(z) = k and k ± = 0. Indeed, there exists δ 1 such that z ∈ S and 0 < | z − z0 | < δ1

implies | g(z) − k|

< 21 |k|

which in turn tells us that z ∈ S and 0

We now ﬁnd δ 2 such that

< |z − z0| < δ1

z ∈ S and 0 < | z − z0 | < δ2

Now, if δ = min(δ 1, δ2 ),

z ∈ S and 0

implies |g(z)|

> 21 |k|

implies | g(z) − k|

< 21 |k|2ε

< |z − z0| < δ implies ² 1 ²² |k − g(z)| < |g(z)k| < 21 |k|2 ε/( 21 |k|2) = ε k²

² ² 1 ² ² g(z) −

The real and imaginary parts of a complex function, f (z)

= re f (z) + i im f (z)

may be considered separately. If lim f (z) = l

z→z 0

= α + iβ (α, β ∈ R)

(2.4)

30

Topology of the Complex Plane

then, because

|re f (z) − α| = |re( f (z) − l)| ≤ |( f (z) − l)| we deduce from the deﬁnition that lim re f (z)

(2.5)

lim im f (z)

(2.6)

z →z 0

Proposition 2.8 (ii) now implies that z→z0

Conversely, if (2.5,2.6) both hold, Proposition 2.8 (i) implies (2.5). We can rephrase the preceding argument by recalling that if S ⊆ R2 the limit of a real function φ : S → R of two variables, φ(x, y) for (x, y) ∈ S is deﬁned as follows:

φ (x, y) → λ as (x, y) →³ (a, b) if, given ε > 0, there exists δ > 0 such that for all (x, y) ∈ S, 0 < (x − a)2 + (y − b)2 < δ implies |φ (x, y) − λ| < ε Identifying (x, y) with z = x + iy ∈ C, and setting z0 = a + ib, this may be written as z ∈ S and 0 < | z − z0 | < δ implies |φ(x, y) − λ| < ε If we now write f (z)

= u(x, y) + iv(x, y)

where the real and imaginary parts of f are considered as real functions u and v of two real variables x, y, we have proved: P RO P O S I T I O N

= a + ib. Then limz→z f (z) = l = α + iβ if and only if u(x, y) → α v(x, y) → β as (x, y) → (a, b)

2.9. Let z0

0

In this way, the limit of a complex functions is equivalent to the limit of a pair of real functions of two real variables. However, the notation in the complex case is generally simpler.

2.3

Continuity The deﬁnition of continuity for a complex function mimics that for a real function, using the complex version of the modulus: 2.10. A function f : S → C is continuous at z 0 ∈ S if, given ε 0 such that

D E FI N I T I O N

exists δ

>

for all z ∈ S, |z − z0|

< δ implies | f (z) − f (z0) | < ε A function is continuous if it is continuous at every point z0 ∈ S.

> 0, there

2.3 Continuity

31

If z0 is a limit point of S, this is equivalent to saying that limz→z 0 f (z) exists and lim f (z)

z →z 0

= f (z0 )

If z0 is an isolated point of S then there is a neighbourhood Nδ (z0) that contains no other points of S apart from z0, so for all z ∈ S, |z − z0|

< δ implies z = z0

which in turn implies

| f (z) − f (z0)| = 0 So a complex function is always continuous at an isolated point, according to the deﬁnition. Actually, this will not be an issue for us, because we will consider only functions where all points of S are limit points. In fact, S is usually open. But it seemed worth tidying up a potential loose end. We can rephrase the deﬁnition of continuity in terms of open discs: f is continuous at z0 ∈ S if, given ε > 0, there exists δ > 0 such that for all z ∈ S, z ∈ Nδ (z0 ) implies f (z) ∈ Nε ( f (z0)) Or, more succinctly, f (N δ (z0))

⊆ Nε ( f (z0 ))

We can develop an alternative way to deﬁne continuous functions in terms of open sets. First we need a generalisation: a subset V ⊆ S is said to be relatively open in S, or just open in S for short, if for every z0 ∈ V there exists σ > 0 such that Nσ (z0 ) ∩ S ⊆ V .

={ ∈ ∈ + σ/ ±∈

Example 2.11. A relatively open set need not be open. The interval (a, b) x R: a x b is open in R, but not in C. In fact, when S C let x (a, b) and suppose that 0. Then x i 2 Nσ (x) Nσ (x) S, but x i 2 V . See Figure 2.5.

+ σ/ ∈

Figure 2.5 The interval (a, b) is open in

=

R but not in C.

=

32

Topology of the Complex Plane

Using the standard set-theoretic notation f −1 (U) = {z ∈ S : f (z)

∈ U}

we get an alternative characterisation of a continuous function: 2.12. A complex function f : S → every open set U in C, the set f −1 (U) is open in S.

P RO P O S I T I O N

C is continuous if and only if, for

Proof. Suppose that f is continuous and U is open. Let z0 ∈ f −1(U). Then f (z0 ) ∈ U so there exists ε > 0 such that Nε ( f (z0 )) ⊆ U. By continuity of f there exists δ > 0 such that f (Nδ (z0 ) ∩ S) ⊆ Nε ( f (z0)) ⊆ U

Hence

Nδ (z0 ) ∩ S ⊆ f −1 (U)

and f −1 (U) is open. Conversely, suppose that f −1(U) is open in S for every open set U. Given z0 ∈ S and ε > 0, the set Nε ( f (z0)) is open, so f −1 (Nε ( f (z0 ))) is open in S and there exists δ > 0 such that Nδ (z0 ) ∩ S ⊆ f −1 (Nε ( f (z0 )))

Hence

f (Nδ (z0 ) ∩ S) ⊆ Nε ( f (z0))

so f is continuous. Proposition 2.12 is favoured by topologists as the deﬁnition of continuity. It is particularly useful when S is itself an open set. Then, given a subset V ⊆ S that is open in S, and any z0 ∈ V, there exist σ , ε > 0 such that Nσ (z0 ) ⊆ V

(because V is open in S)

Nε (z0 )

(because S is open)

⊆S

Taking δ = min(ε, σ ), we deduce that Nδ (z0 ) ⊆ V so V is an open set in C. This proves: 2.13. If S ⊆ C is open, then f : S → every open set U, the inverse image f −1(U) is open.

C O RO L L A RY

C is continuous if and only if, for

Figure 2.6 illustrates this result. Composition of two functions is deﬁned in the usual manner: if f : S g : T → C where f (S) ⊆ T, then g ◦ f : S → C is given by g ◦ f (z)

= g( f (z))

→ C and

2.3 Continuity

Figure 2.6 Deﬁnition of continuity when S is open in

P RO P O S I T I O N 2.14. If f is continuous at z0 is continuous at z0 .

33

C.

∈ S and g is continuous at f (z0), then g ◦ f

Proof. Easy exercise. Addition, subtraction, multiplication, and division of complex functions f1 : S and f2 : S → C are deﬁned by f 1 + f2 : S → C where ( f 1 + f2 )(z) = f1 (z) + f2(z)

f 1 − f2 : S → C where ( f 1 − f2 )(z) = f1 (z) − f2(z) f1 f2 : S → C where ( f 1f2 )(z) = f 1(z)f 2(z) (z

f1 /f2 : S → C where ( f 1/f 2)(z)

= f1(z)/ f2(z)

→C

(z ∈ S)

(z ∈ S)

∈ S) (z ∈ S² )

Here S² = {z ∈ S : f 2(z) ± = 0}. Warning: the composition g ◦ f is often abbreviated to g f when it is clear that the product is not intended. 2.15. If f1 and f 2 are continuous, then so are f1 f1 /f2 (when f2(z) ± = 0). P RO P O S I T I O N

+ f2 , f 1 − f2, f1 f2 , and

Proof. This is a direct consequence of Proposition 2.8. This result lets us show very quickly that certain functions built up from functions known to be continuous are themselves continuous. For instance, the constant function k(z) = c (where c ∈ C) and the identity function I(z) = z are clearly continuous. We give the easy proofs. Let ε > 0. For k take any δ > 0; then

|z − z0| < δ implies |k(z) − k(z0 )| = |c − c| = 0 < ε

For I take δ = ε; then

|z − z0 | < ε implies |I(z) − I(z0)| = | z − z0 | < ε

34

Topology of the Complex Plane

On that basis, Proposition 2.15 shows immediately that f (z) = cz is continuous. By induction on n, it follows that f (z) = czn is continuous for any n ∈ N. A further induction proves that any polynomial function f (z) = anzn + an−1zn−1 + · · · + a0

is continuous. Then any rational function p(z)/q(z), where p, q are polynomials, is continuous at any point z0 where q(z0) ± = 0. Using these ideas we seldom have to perform any intricate epsilon-delta calculations. Example 2.16. The modulus function m(z) take . Then

δ=ε

= |z| is continuous for all z. Given ε > 0,

|m(z) − m(z0)| < ε implies ||z| − |z0|| ≤ |z − z0| < ε

using (1.12).

| z| +17z +1066z is continuous for z 1. Proving this with an Example 2.17. f (z) 1+z explicit choice of for given is tedious and messy. A quicker way is to observe that z2 z z and is therefore continuous, 17z3 1066z is a polynomial and so is also continuous, and the same goes for z 1. Since we are dividing by this, we must have z 1.

| | = | || |

δ

=

2

±= −

3

ε

+

+

±= −

Writing f (z) = u(x, y) + iv(x, y) for z = x + iy ∈ S, where u, v are real-valued functions of the real variables x, y, Proposition 2.9 gives: The function f (z) = u(x, y) + iv(x, y) is continuous at z 0 = x0 + iy0 if and only if u, v are continuous at (x0 , y0 ).

P RO P O S I T I O N 2.18.

=

= +

= − + ∈

= −

Example 2.19. If f (z) z2 , then f (z) (x iy)2 x2 y2 2ixy. Hence u(x, y) x2 y2 and v(x, y) 2xy. The function f is continuous for all z C, just as u, v are continuous functions of x, y for all (x, y) R2 .

=

An interesting case occurs when S is the real interval S = [a, b] = {x ∈ R : a ≤ x ≤ b}

considered as a subset of C. Here z = x + 0i, so we can simplify notation and write f (z) = u(x) + iv(x). Now the function f : [a, b] → C is continuous if and only if both u and v are continuous.

Example 2.20. f : [0, 1] C, where f (x) 3 and v(x) x are continuous.

=

= x2 + ix3 is continuous, since both u(x) = x2

Functions deﬁned on a real interval play a central role because they deﬁne paths, as we now discuss.

2.4 Paths

2.4

35

Paths In the early development of complex analysis, mathematicians worked out how to integrate complex functions along paths in C, without worrying too much about the meaning of ‘path’. This happened back in the days when notions of continuity and limits were still under development. The intuitive idea of a path (or curve) was something that could be drawn by moving a pencil (with an inﬁnitely ﬁne point) without making any jumps. Such paths were assumed to have various nice properties, such as having a continuously varying tangent, or not crossing themselves. These properties were not made explicit until the foundational issues in analysis were sorted out, in particular by Bolzano and Weierstrass. Later it turned out that some of these assumptions are not necessary to develop a satisfactory theory of integration, while others must be made precise to avoid running into contradictions. We have no wish to make readers repeat all the twists and turns of history; as we have said, some battles have already been won. But from time to time we ﬁnd it useful to point out some of the pitfalls that can arise, and to emphasise some subtle distinctions – for example, the deﬁnition of a path as a map γ from a real interval [a, b] into the complex plane, and the distinction between this map and its image. Our pictures generally show the image, leaving the reader to infer the map. We often add an arrow as a reminder that as a parameter t runs through the interval [a, b], the point γ (t) moves along the image in a speciﬁc direction. One of the pleasant features of complex analysis is that fairly often that information is all we need. At various points in the book we also ﬁnd it useful to impose further conditions on paths, to ensure that they have appropriate properties. The upshot of more than a century of deep contemplation and debate is the following deﬁnition: D E FI N I T I O N 2.21.

A path in the complex plane is a continuous function

γ : [a, b] → C

(a ≤ b ∈ R)

Its initial point is γ (a) and its ﬁnal point (or end point) is γ (b).

Sometimes we speak of γ as ‘a path in C from z1 to z2 ’ when z1 = γ (a) is the initial point and z2 = γ (b) is the ﬁnal point. When t ∈ [a, b] we also refer to z = γ (t) as a ‘point on the path γ ’, although strictly speaking z is on the image of the function γ . If we think of t as ‘time’ and imagine t increasing from a to b, the point γ (t) traces a curve in the plane from γ (a) to γ (b). When drawing a diagram we often indicate the direction of motion (increasing t) by an arrow, as in Figure 2.7. However, it must be emphasised that this is a makeshift device since the curve may cross itself or turn back on itself, so the picture may get quite complicated. We pick up this point again in Section 6.7.

2.4.1

Standard Paths To simplify matters, when we speak of line segments and circles, considered as paths, we assume they are speciﬁed by the following standard functions:

36

Topology of the Complex Plane

Figure 2.7 A path in

C, showing its parametrisation by t ∈ [a, b].

(i) The line segment L

= [z1, z2] from z1 to z2 is L(t)

= (1 − t)z1 + tz2

(t ∈ [0, 1])

When z1 = a, z2 = b ∈ R then the image of the path is the closed interval [a, b] R ⊆ C. (ii) The unit circle C: C(t)

= cos t + i sin t

(iii) The circle S with centre z0 and radius r S(t) See Figure 2.8.

Figure 2.8 Three standard paths.

(t ∈ [0, 2π ])

≥ 0:

= z0 + r(cos t + i sin t)

(t

∈ [0, 2π ])

2.4 Paths

37

→ C. Left: A path that travels once along the image in a ﬁxed direction. Right: A path that travels once along the image, turns round and travels back, then turns again and travels along the image for a third time.

Figure 2.9 ‘Graphs’ of paths considered as functions [a, b]

2.4.2

Visualising Paths We are used to visualising a function f : R → R using the corresponding graph y = f (x) in the plane. A similar idea can be exploited to visualise paths in a more explicit way. Figure 2.9 shows analogues of graphs for two maps γ : [a, b] → C. The arrows join representative points t ∈ [a, b] to their images γ (t) ∈ C. The left-hand ﬁgure illustrates a path where γ (t) travels along the image without turning back on itself. The right-hand ﬁgure illustrates a path where γ (t) travels along the image to the far end, turns back on itself to revisit the start, then turns back again to travel along the image for a third time. Although paths like this are seldom encountered in practice, they are allowed by the deﬁnition, so we have to take that possibility into account. Example 6.17 below shows that simple formulas can produce this type of behaviour. An alternative would be to change the deﬁnition to rule out such behaviour, but most of the theory works in the general case with no extra effort, so it turns out to be simpler to leave the deﬁnition as it stands, while bearing in mind that simple pictures might be misleading in some respects.

2.4.3

The Image of a Path To emphasise the distinction between a path and its image we introduce a further deﬁnition: A curve C from z1 to z2 in the complex plane is a subset of C that is equal to the image of a path γ : [a, b] → C, where γ (a) = z1 and γ (b) = z2. Given a curve C, a parametrisation of C is a path σ : [c, d] → C such that C is the image of σ , where σ (a) = z1 and σ (b) = z2. The corresponding parameter is a variable t ∈ [a, b]. We sometimes call [a, b] the parametric interval of σ . D E FI N I T I O N 2.22.

38

Topology of the Complex Plane

γ γ

Figure 2.10 The two distinct paths 1 , 2 have the same image, so they deﬁne the same curve.

If we think of t as ‘time’, and imagine a particular parametrisation γ : [a, b] → C, then as t increases from a to b the point γ (t) traces the corresponding curve in the plane from γ (a) to γ (b), moving continuously with t. Example 2.23. The same curve may be traced by many different paths. For example, in Figure 2.10 the paths

γ1(t) = 2(t + it) γ2(t) = t 2 + it 2

(0 ≤ t ≤ 21 )

(0 ≤ t

≤ 1)

traverse the same curve

{x + iy ∈ C : x = y, 0 ≤ x ≤ 1} 2.5

Change of Parameter Deﬁnition 2.21 equips every path γ : [a, b] → C with a parameter t ∈ [a, b]. It is often convenient to change the parameter and its parametric interval, because suitable choices can simplify calculations and proofs. In this section we discuss such changes. We begin with Example 2.23, which distinguishes a path from its image by exhibiting two different paths that have the same image curve. Dynamically, these two paths trace the same curve in the same direction, but at different speeds. They are related by the change of parameter ρ : [0, 21 ] → [0, 1],√where ρ (t) = 21 t2 . This has an inverse map ρ−1 : [0, 1] → [0, 21 ] given by ρ−1 (s) = 2s. We now discuss general changes of parameter, deﬁned as follows: 2.24. Let γ : [a, b] is continuous and satisﬁes

D E FI N I T I O N

→ C be a path, and suppose that ρ : [c, d] → [a, b]

ρ(c) = a

ρ (d) = b

2.6 Subpaths and Sums of Paths

so in particular ρ is onto [a, b]. Then the composition γ ◦ ρ : [c, d] → We call γ ◦ ρ a reparametrisation of γ . Note that the parametric interval changes when [c, d] ± = [a, b].

2.5.1

39

C is also a path.

Preserving Direction If we impose extra conditions on ρ , a change of parameter can preserve extra properties. The image of the path is important, but another property is also vital: the direction in which the path is traced. Now our mental image involves the dynamics of a point moving along the path, as well as the image that it traces out. At this stage we are working with general continuous paths, so we adopt the following approach. Recall that a map σ : [a, b] → [c, d] is strictly increasing if a ≤ t 1 < t2 ≤ b implies σ (t1) < σ (t2). Both of the above maps ρ and ρ −1 are strictly increasing. In real analysis it is proved that any strictly increasing continuous map has a strictly increasing continuous inverse. We give the easy proof for completeness: L E M M A 2.25. If ρ : [a, b] → [c, d] is continuous and strictly increasing with ρ (a) = c and ρ (b) = d, then ρ has a strictly increasing continuous inverse ρ −1 : [c, d] → [a, b].

Proof. If s ∈ [c, d] then ρ (a) ≤ s ≤ ρ (b). By the Mean Value Theorem there exists t ∈ [a, b] such that ρ (t) = s. This determines a strictly increasing inverse function ρ−1 : [c, d] → [a, b], where ρ and ρ−1 both map open intervals to open intervals, so ρ−1 is also continuous. We can now state: 2.26. Let C be a curve, parametrised by two maps γ : [a, b] → C and : [c, d] → C. The maps have the same direction if there is a strictly increasing function : [a, b] → [c, d] such that

D E FI N I T I O N

λ ρ

γ =λ◦ρ

Then ρ is a direction preserving change of parameter.

If the change in parameter ρ is not increasing, it can lead to part of the curve being traced back and forth as t increases. In Figure 2.11, as t increases from p to q, the value of ρ (t) increases, then decreases, then increases again. As this happens, the points on the curve from P to Q are traversed ﬁrst in the direction from P to Q, then back to P, then from P to Q once more. The curve is the same, but the distance travelled increases. This means that when we calculate the length of a curve, deﬁned in Section 6.3, we must prescribe how the curve is traced.

2.6

Subpaths and Sums of Paths In the general theory, and in applications, it is useful to be able to chop paths into pieces, or join pieces together.

40

Topology of the Complex Plane

Q z-plane

image( γρ)

p z = γ (ρ (t))

d

ρ (t)

γ c

a t

d

t

q b

ρ (t) changes direction as t increased from p to q

ρ a

p

c

ρ (t)

b

Figure 2.11 Left: A change in parameter. Right: If

changes direction along the image curve.

ρ has a graph like this then the parameter t

2.27. For an arbitrary path γ : [a, b] → C, a subpath is obtained by restricting γ to a subinterval [c, d], where a ≤ c ≤ d ≤ b. If

D E FI N I T I O N

a = x0

< x1 < · · · < xn = b and γr is the subpath obtained by restricting γ to [xr −1 , xr ], we write γ = γ1 + · · · + γn In so doing, we think of γ as being made up by tracing along the paths γ1 , . . . , γn taken in order, see Figure 2.12.

Figure 2.12 Decomposing a path into subpaths.

Similarly, if the ﬁnal point of a path γ1 coincides with the initial point of a path γ2 , it is useful on occasion to create a combined path by ﬁrst tracing γ1 and then γ2. There is a minor technicality here, because γ1(b) can be the same as γ2 (c) even though γ1 is deﬁned on [a, b] and γ2 deﬁned on [c, d] where b ± = c. The next example shows how to get round this difﬁculty.

2.6 Subpaths and Sums of Paths

41

Figure 2.13 Joining two paths by shifting parameters.

Example 2.28. Suppose that

γ1(t) = t (t ∈ [0, 1]) γ2(t) = cos t + i sin t (t ∈ [0, π ]) as in Figure 2.13. Here γ1 (1) = γ2 (0) but 1 ± = 0. In such a case we extend our notation a little. Suppose that γ1 : [a, b] → C, and γ2 : [c, d] → C, with γ1 (b) = γ2(c). Then we deﬁne the combination γ = γ1 + γ2 : [a, b + d − c] → C by ± (t ∈ [a, b]) γ (t) = γγ1(t) (t + c − b) (t ∈ [b, b + d − c]) 2 In effect, what we have done is to shift the parametric interval of the second path from [c, d] to [b, b + d − c], by adding b − c to all points in [c, d]. In Example 2.28, for instance,

γ (t) =

±

t cos(t − 1) + i sin(t − 1)

(t ∈ [0, 1]) (t ∈ [1, 1 + π ])

and γ consists of the line segment γ1 followed by the semicircle γ2. The path γ1 + γ2 is deﬁned only when the ﬁnal point of γ1 is the same as the initial point of γ2, so perhaps this is not a fully appropriate use of the + sign. However, we do have (γ1 + γ2 ) + γ3

= γ1 + (γ2 + γ3 )

(2.7)

whenever the appropriate end points coincide, so we can omit the parentheses in such a ‘sum’, and more generally write γ1 + · · · + γn to indicate the path obtained by successively tracing γ1, . . . , γn, whenever the ﬁnal point of each γr−1 coincides with the initial point of γr (1 ≤ r ≤ n).

42

Topology of the Complex Plane

Equation (2.7) states that addition of paths is associative, but in general it is not commutative. Exercises 7 and 8 provide simple examples to show that when γ1 + γ2 is deﬁned, γ2 + γ1 may not be, and even when it is, γ1 + γ2 need not equal γ2 + γ1. For a path γ : [a, b] → C, another useful concept is the opposite path −γ : [a, b] → C deﬁned by

−γ (t) = γ (a + b − t) (t ∈ [a, b]) As t increases from a to b, so −γ describes the same curve as γ , but it does so in the reverse direction. If γ is a path from z1 to z2, then −γ is a path from z2 to z1 . Example 2.29. Suppose that L is the line segment [z1, z2] given by

Then −L is given by

L(t) = (1 − t)z1 + tz 2

−L(t) = tz 1 + (1 − t)z2

(t

∈ [0, 1])

(t

∈ [0, 1])

which is, of course, [z2, z1 ]. If S is the circle centre z0 and radius r ≥ 0, then S(t) = r(cos t + i sin t) (t

∈ [0, 2π ])

describes the circle once, anticlockwise, while the opposite path

−S(t) = z0 + r(cos(2π − t) + i sin(2π − t)) = z0 + r(cos t − i sin t) (t ∈ [0, 2π ])

(t

∈ [0, 2π ])

describes it once, clockwise. In a sum such as γ1 + (−γ2 ) + γ3 we omit the parentheses and write γ1 − γ2 + γ3 . This is the path that traces ﬁrst along γ1, then back along the opposite path to γ2 , and then along γ3. As always, the appropriate ﬁnal and initial points must agree. See Figure 2.14.

γ − γ2 + γ3 .

Figure 2.14 The path 1

2.7 The Paving Lemma

Figure 2.15 The path CR

43

+ L1 − Cε + L2.

< ε < R and CR (t) = R(cos t + i sin t) (t ∈ [0, π ]) Cε (t) = ε(cos t + i sin t) (t ∈ [0, π ]) L1 (t) = t (t ∈ [−R, −ε ]) L2 (t) = t (t ∈ [ε, R])

Example 2.30. Suppose that 0

Then CR + L1 − Cε + L 2 is the path describing the curve in Figure 2.15 once, anticlockwise.

2.7

The Paving Lemma We now come to a technical lemma that plays a vital role in our approach to complex integration, which we call the Paving Lemma. This result is so important that we present two different proofs of it. To set it up, we ﬁrst need another piece of terminology. A path γ : [a, b] → C is said to be a path in S if the image

{γ (t) : t ∈ [a, b]} ⊆ S We signify this by writing γ : [a, b] → S. =

+

π

Example 2.31. The unit circle C(t) cos t i sin t (t [0, 2 ]) is a path in the region z 2 , Figure 2.16. S z C : 21 A region of this form, contained between two concentric circles, is called an annulus.

={ ∈

≤| |≤ }

This example is very simple. It must not lull us into a false sense of security: far more intricate paths are possible.

44

Topology of the Complex Plane

Figure 2.16 Path in an annulus.

Figure 2.17 A more complicated path.

= {z ∈ C : z ±= 1/n for a non-zero integer n} and deﬁne ± σ (t) = 0t + it cos(π/t) (t(t =∈ [0)−1, 1] \ {0}) Then σ is a path in V , Figure 2.17. Here σ winds in between the points 1/ n where n is a non-zero integer, crossing the real axis when t = 1/(n + 21 ). Example 2.32. Let V

It does not take much imagination to realise that an open set S can be very complicated, and a path γ can be intertwined in S in a very intricate manner. The space-ﬁlling curve described in Section 2.9 is given by a path whose image is the unit square, including its interior. Much more complicated images are possible. Rather than trying to imagine all possible intricacies, we sidestep the issue, as follows. If S is an open set,

2.7 The Paving Lemma

45

Figure 2.18 Paving a path with a sequence of discs. But could this process get stuck?

we can cover γ with a ﬁnite number of open discs, all contained in S. Then we use the discs to simplify the path, as the next result explains. 2.33 (Paving Lemma). Let γ : [a, b] → S be a path in an open set S. Then there exists a subdivision a = t0 < t 1 < · · · < tn = b such that each subpath γr obtained by restricting γ to [tr −1, tr ] lies inside an open disc Dr ⊆ S. LEMMA

Proof. Because S is open, there is an open disc D1 such that γ (a) ∈ D1 ⊆ S. The idea of the proof is that if the path has been paved up to some point, we can always add another disc to take the paving further, unless we have reached the far end. Making this argument rigorous takes a little care, however, as we now indicate. The disc D1 is the ﬁrst in a sequence of discs, obtained by moving along the path, covering it bit by bit with discs as in Figure 2.18. The existence of D1 shows we can get started. Indeed, since γ is continuous, there exists δ > 0 such that

γ (t) ∈ D1 whenever a ≤ t < a + δ

(2.8)

Our only fear is that we might not progress all the way; for instance, the discs might decrease dramatically in size and we might never reach the end. Some real analysis dispels this fear. If a ≤ x ≤ b, say that γ restricted to [a, x] ‘can be paved’ if it can be subdivided into a ﬁnite number of subpaths γ1 , . . . , γm , where each γr lies inside an open disc Dr ⊆ S. Let P be the set of all x ∈ [a, b] such that [a, x] can be paved. We know that a ∈ P so P is non-empty. Also P is bounded above by b. Therefore, by real analysis, P has a least upper bound c, where a ≤ c ≤ b. We claim that c = b, which will prove the lemma. For a contradiction, suppose that c < b. Since S is open, there exists an open disc D such that c ∈ D ⊆ S. By continuity of γ , there exists ε > 0 such that if c −ε < t < c +ε and t ∈ [a, b], the image γ (t) ∈ D. By (2.8), a + δ/2 ∈ P, so c > a. Since c is the least upper bound of P, there exists x ∈ P with c−ε < x < c. Therefore the segment {γ (t) : a ≤ t ≤ x} can be paved by open

46

Topology of the Complex Plane

discs D1 , . . . , Dm all lying inside S. Let Dm+1 = D. Then the segment {γ (t) : a ≤ t ≤ c} can be paved by open discs D1 , . . . , Dm , Dm +1 all lying inside S. Finally, we claim that c = b. If c < b, continuity of γ implies that γ (c + t) ∈ D for all t such that 0 < t < τ , therefore [a, c + τ/2] can be paved, contradicting c being an upper bound for P. Therefore c = b so [a, b] can be paved as required. For future reference, and because this result is used repeatedly, we now give a second proof, which in some respects is simpler. A generalisation to two dimensions will also prove useful in Proposition 9.5. Alternative Proof of Lemma 2.33. We show more: the subdivision can be obtained by repeatedly bisecting [a, b] a ﬁnite number of times. For the purposes of this proof only, it is useful to deﬁne and interval to be pavable if it can be bisected repeatedly a ﬁnite number of times into closed intervals, so that the image of every interval obtained in this manner lies inside a disc in D. Otherwise, the interval is unpavable. Suppose the result is false. Then [a, b] is unpavable. Bisect [a, b] to get two closed intervals, intersecting at the midpoint. At least one these intervals, call it I1 , is unpavable. Otherwise we could pave a ﬁnitely repeated bisection of each interval by ﬁnitely many discs, whose union paves [a, b]. Bisect I 1. The same argument applies to the two halves, so we get an unpavable closed interval I2 ⊆ I1 of half the size. Because we are assuming the result false, this sequence of bisections continues indeﬁnitely to give a nested sequence of unpavable closed intervals [a, b] ⊇ I1

⊇ I2 ⊇ · · ·

each half the size of the previous one. The intersection of these intervals is easily seen to be a unique point z0 ∈ D (Cauchy sequences of real numbers converge). However, D is open, so there is a disc Nε (γ (z0)) ⊆ D. Since γ is continuous, γ −1(Nε (γ (z0))) is open in [a, b] and contains z0. Therefore it contains I n for sufﬁciently large n. Now In ⊆ Nε (γ (z0 )), so it is pavable (with one disc), a contradiction. In Example 2.31, for any point on the unit circle C, we have N1/2 (z) ⊆ S. Clearly a ﬁnite number of such discs will pave C within S, Figure 2.19. However, the path in Example 2.32 cannot be paved by a ﬁnite number of discs in V . Any disc containing the origin (which lies on σ includes points of the form 1/n lying outside V . But this also means that V is not open. So the conditions required for the Paving Lemma are not satisﬁed. Readers familiar with basic point-set topology will recognise that the proof is ‘really’ about the compactness of the closed interval [a, b]. We prefer not to develop the abstract machinery of compactness here; the Paving Lemma performs the same task.

2.8

Connectedness We now deﬁne an important topological property, which is central to the theory of complex functions.

2.8 Connectedness

47

Figure 2.19 Paving a circular path inside an annulus is easy.

Figure 2.20 A disc is path-connected.

D E FI N I T I O N 2.34. A subset S ⊆ a path γ in S from z1 to z2 .

C is path-connected if, given z1 , z2 ∈ S, there exists

Example 2.35. Any open disc N r (z0 ) is path-connected. For, given z1 , z2 L(t) (1 t)z1 tz2, (t [0, 1]). Then

= −

+

∈ |L(t) − z0 )| = |(1 − t)z1 + tz2 − z0 | = |(1 − t)z1 − (1 − t)z0 + tz2 − tz 0| ≤ (1 − t)|z1 − z0 | + t|z2 − z0 | < (1 − t)r + tr = r

∈ Nr (z0 ), let

so the line segment L lies in Nr (z0 ). See Figure 2.20.

={ ∈ ≤

= + it2 , t ∈ R} is path-connected. For if t 1 + it21 γ = t + it2 (t ∈ [t1 , t2 ]) is a path in S between

Example 2.36. The set S z C:z t 2 and t2 it2 S where t1 t2 , then (t) those points. See Figure 2.21.

+

48

Topology of the Complex Plane

{ ∈ C : z = t + it2 , t ∈ R} is path-connected.

Figure 2.21 The set z

Figure 2.22 A step path.

It is often convenient to use a very simple type of path. A step path in S is a path γ : [a, b] → S together with a subdivision a = t0 < t1 < · · · < tn = b such that on each subinterval [tr −1 , tr ] either re γ or im γ is constant. This means that the image of γ consists of a ﬁnite number of straight line segments, each parallel to the real or imaginary axis, see Figure 2.22. In other words, γ is a step path if γ = γ1 + · · · + γn where each γr is a line segment parallel to the real or imaginary axis. A subset S ⊆ C is step-connected if, given z1 , z2 ∈ S, there exists a step path γ in S from z1 to z2 .

D E FI N I T I O N 2.37.

Evidently a step-connected path is path-connected. The converse, however, is not always true: Example 2.36 is path-connected but not step-connected. A path as simple as the diagonal line segment [0, 1 + i] is path-connected but not step-connected. The main issue here is that unlike a general continuous path, a step path consists of ﬁnitely many

2.8 Connectedness

49

straight line segments. But the requirement that these segments should be horizontal or vertical also comes into play. These considerations are mentioned here to help develop intuition, but they do not cause problems for the theory, because every path-connected open set is step-connected. This constitutes the ﬁrst success – albeit a minor one – for the Paving Lemma. To prove this, we ﬁrst observe a simple fact: LEMMA

2.38. Any open disc is step-connected.

Proof. Consider an open disc S = Nr (z0). Changing coordinates to z − z0 (translating the origin) we may assume without loss of generality that z0 = 0. Let z1 = x1 + iy1 ∈ S. We show that there is a step path in S from z1 to the centre 0. Let L1 be the (vertical) line segment from z1 to x1 , and L 2 be the (horizontal) line segment from x1 to 0. Then simple inequalities show that L 1 and L 2 are inside S, and L 1 + L 2 is a step path from z1 to 0. By the same argument, if z2 ∈ S there is a step path L3 + L4 in S from z2 to 0. But now L1 + L 2 − L 4 − L3 is a step path in S from z1 to z2. We can now prove: P RO P O S I T I O N 2.39.

A path-connected open set is step-connected.

Proof. Let z1, z2 in S, which is open, and let γ be a path from z1 to z2 in S. By the Paving Lemma there is a ﬁnite sequence of open discs D1, . . . , Dr ⊆ S and a path γr in each Dr such that γ = γ1 + · · · + γr . By Lemma 2.38 there is a step path σ r in each Dr with the same initial and ﬁnal points as γr . Now σ = σ1 + · · · + σr is a step path in S from z1 to z2 . On the other hand, the parabola S in Example 2.36 is path-connected, but not stepconnected. This does not conﬂict with the proposition above, because S is not open in C. Arbitrary sets need not be path-connected. However, for any set S we can deﬁne the relation z1 ∼ z2 (z1, z2 ∈ S) to mean that there is a path in S from z1 to z2 . It is easy to see that ∼ is an equivalence relation (Exercise 7 below) and that each equivalence class is path-connected. These equivalence classes are called the path-connected components, or for simplicity the connected components or just components of S. (We always take S open in the sequel, so here is no conﬂict with the more general notions of connectedness and connected components that are standard in point-set topology, but irrelevant here.) For instance, the connected components of A are clearly A1

= {z ∈ C : |z| ± = 1}

= {z ∈ C : |z| < 1}

A2

= {z ∈ C : |z| > 1}

If S is open then its connected components are all open: given z0 in a connected component S0 of S, since some Nr (z0) ⊆ S and Nr (z0) is path-connected, we must have Nr (z0) ⊆ S0 , so S0 is open.

50

Topology of the Complex Plane

A particularly interesting case is the complement of the image of a path

C, or more brieﬂy the complement of a path, by which we mean {z ∈ C : z ±= γ (t) for any t ∈ [a, b]}

γ : [a, b] →

The complement may be connected, for instance when

γ (t) = t

(t

∈ [0, 1])

On the other hand, it may have two or more components, Figure 2.23. Indeed, the complement of γ = γ1 − γ2 , where

γ1 (0) = 0, γ1 (t) = t + it sin(π/t) γ2 (0) = 0, γ2 (t) = t − it sin(π/t)

(0 < t

≤ 1) (0 < t ≤ 1)

has inﬁnitely many components, Figure 2.24. 2.40. A subset S ⊆ C is bounded if there exists K for all z ∈ S. A subset S ⊆ C is unbounded if it is not bounded.

D E FI N I T I O N

≥ 0 such that |z| ≤ K

Figure 2.23 Paths whose complements have one, two, or three components.

Figure 2.24 A path whose complement has inﬁnitely many components.

2.8 Connectedness

51

However complicated a path may be, we can prove: P RO P O S I T I O N 2.41. (i) The image of a path in C is closed and bounded. (ii) The complement of a path in C is open, and precisely one component is unbounded.

Proof. For the second part of (i), let the path be γ : [a, b] → C, and let m(t) = |γ (t) |. Since it is the composition of two continuous functions, m : [a, b] → R is continuous. By real analysis it is bounded, say m(t) ≤ K, whence |γ (t)| ≤ K for all t ∈ [a, b] and the image of [a, b] lies inside the disc centre 0 of radius K. The ﬁrst part of (i) follows from (ii). To prove (ii), suppose that z0 lies in the complement T of γ . Then

µ(t) = |γ (t) − z0 | is a positive real number for all t ∈ [a, b]. Further, µ is continuous, so it is bounded and attains its bounds. That is, there exists k ≥ 0 such that µ(t) ≥ k for all t ∈ [a, b], and there exists t0 ∈ [a, b] such that µ (t0 ) = k. If k = 0 then z0 ∈ γ ([a, b]), but it is in T . So k > 0. Then Nk (z0) ⊆ T , so T is open. Finally, the image of γ lies inside A = {z ∈ C : |z| ≤ K } and the complement

B = {z ∈ C : |z| > K }

is clearly connected. Therefore any unbounded component of T intersects B, and since components are disjoint, at most one of them can do so. Further, B lies inside some component, so there is exactly one unbounded component. 2.42. A closed, bounded set in Rn, in particular one in C = R2 , is said to be compact. Compactness is a very important property in point-set topology (where the deﬁnition is given in a more general form). In this text we do not need to set up the full machinery of compactness, because the simple cases that arise can be tackled bare-handedly using real analysis – as we did in proving Proposition 2.41. REMARK

We now deﬁne a special type of set that is fundamental to the theory in this text. 2.43. A domain is a non-empty, path-connected, open subset of the complex plane. D E FI N I T I O N

By Proposition 2.39, a domain is also step-connected, and this property will be useful later. This means that we can refer to a domain as being ‘connected’, meaning either path- or step-connected as appropriate. As the theory of complex analysis unfolds, the reader will see the immense importance of this deﬁnition. We restrict the concept of a complex function f to those of the form f : D → C where D is a domain. Openness of D lets us deal neatly with limits, continuity, and differentiability, because z0 ∈ D implies that Nε (z0) ⊆ D for some ε > 0, so f (z) is deﬁned for all z near z0 . Connectedness of D guarantees there is a (step-) path between any two points in D, which lets us deﬁne the integral of f along such a path.

52

Topology of the Complex Plane

However, restricting complex functions to those deﬁned on domains has far subtler consequences than merely providing a platform for the appropriate deﬁnition. Those who have studied the intricacies of real analysis in depth will ﬁnd untold riches in complex analysis, quite unlike the real case. For instance, if two differentiable complex functions are deﬁned on the same domain D and they happen to be equal on a small disc in D, they are equal throughout D. No such result holds for general differentiable functions in the real case. This result is just one of many that illustrate the beauty and simplicity of complex analysis. We shall not prove it until Chapter 10, but we mention it here to underline the fundamental importance of establishing appropriate topological foundations for the subject.

2.9

Space-ﬁlling Curves A space-ﬁlling curve is deﬁned by a path whose image ﬁlls a two-dimensional region, typically the unit square. Such curves show that our default mental image of a continuous path can be distinctly misleading. To explain why care is needed, we digress to describe one simple construction of such a path. Let U2 be the unit square {x + iy : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1}. 2.44. A space-ﬁlling curve is a continuous path that the image of γ is the whole of U2 .

D E FI N I T I O N

γ

: [0, 1]

→ U2 such

Giuseppe Peano constructed the ﬁrst space-ﬁlling curve in 1890, motivated by Georg Cantor’s proof of 1878 that [0, 1] and U2 have the same transﬁnite cardinal. In 1879 Eugen Netto had proved that there is no continuous bijection between these sets; Peano showed that such a map exists if we do not require it to be bijective. In 1891 Hilbert published another example of a space-ﬁlling curve, and included a picture. Both constructions are fairly complicated: Figure 2.25 shows early stages in their construction. Many others have since been found. Details can easily be found on the Internet, and are given in Sagan [18] and Bader [2]. We give a simpler construction, which at each stage produces a curve whose image is a square grid. Peano’s result was unexpected because it challenges the naive notion of ‘dimension’. A curve has dimension 1, but U2 has dimension 2. It was counterintuitive that a continuous map can increase the dimension of a space. Exercises 15–17 show that there is no bound on the size of this increase. This phenomenon led to more careful theories of the concept of dimension for topological spaces, and the realisation that stronger conditions than continuity are needed for some intuitively plausible properties to hold. The idea behind the construction of such curves is purely topological. It relies on simple properties of uniform convergence, as studied in real analysis courses. It requires the deﬁnition and properties of the closure of a subset S ⊆ C, stated in Deﬁnition 2.5. 2.45. Let n ∈ N. Suppose that:

P RO P O S I T I O N

γn

: [0, 1]

→ U2 be a sequence of continuous paths, for

2.9 Space-ﬁlling Curves

53

Figure 2.25 Left: An early stage in the construction of Peano’s space-ﬁlling curve. Right: An early

stage in the construction of Hilbert’s space-ﬁlling curve.

(i) The sequence of functions (γn) is uniformly convergent. (ii) The closure of the union of the images of all γn is the whole of U2 . Then the limit

γ = nlim →∞ γn

is continuous, and its image is U2 . Before giving the proof, it must be made clear that we are not just asserting that γ comes close to every point of U2 . It actually passes through every point of U2 in the sense that if z ∈ U2 there exists t ∈ [0, 1] such that γ (t) = z. In contrast, the union of the images of the γn is not the whole of U2 . It is just dense in U2 : that is, its closure is U2 . Proof. By condition (i), a basic theorem in real analysis implies that the limit γ exists and is continuous. Let z ∈ U2 . By condition (ii), for each m > 0, m ∈ N, there exists tm ∈ [0, 1] and nm ∈ N such that |γnm (tm ) − z| < m1

The subsequence γnm also tends uniformly to γ . The sequence (tm ) lies in [0, 1] which is closed and bounded, so it has a convergent subsequence, with limit t 0. We claim that γ (t 0) = z. By uniform continuity, if m is large enough,

|γn

m

(t0 ) − γ (t0)|

< m1

So

for all m, so γ (t0 ) = z as claimed.

|γ (t0 ) − z| < m2

54

Topology of the Complex Plane

We now construct such a sequence of maps γn : [0, 1] → U2. We prescribe it geometrically, but making the construction precise is routine. Deﬁne γ0 so that its image is the boundary of U2 described anticlockwise, parameterised proportionally to arc-length; see top left of Figure 2.26. To obtain γ1, replace each segment of γ0 by a T-shaped polygon, as shown at the bottom of Figure 2.26. The extra part of the T always extends to the left of the segment, as oriented by the arrow. The path γ1 goes along the original path to its midpoint, turns left, turns back on itself to return to the midpoint, and then continues along the original segment to its end point. If we do this for each of the four segments of γ0 and add the T-shaped paths (whose ends and starts ﬁt together correctly) we obtain the path γ1 at top right of Figure 2.26. For clarity, the grey polygon shows the direction in which t traverses γ1 . This path visits the centre of the square four times, but all segments are treated in the same manner, making it easier to deﬁne the γn inductively. We scale the arc-length parametrisation so that each γn has the same parametric interval [0, 1]. We now repeat the same construction on every straight segment of γ1 to obtain γ2 , and continue inductively to deﬁne γn for all n ∈ N. The image of γn is a square grid of lines separated by distance 2−n . Clearly the union of these images is dense in U2: it consists of all z = x + iy for which either x or y is of the form m2−n for 0 ≤ m ≤ 2n, where m ∈ N. So condition (i) holds, and we can deﬁne γ as the limit of the γn . By considering the construction step at the bottom of Figure 2.26, it is easy to prove that

|γn(t) − γn+1(t)| ≤

2 · 2−n

from which condition (ii) follows (by summing a geometric series). Therefore γ is a space-ﬁlling curve by Proposition 2.45. Although the image of γn is a grid, the actual structure of γ becomes increasingly complicated because of the way it zigzags back and forth. If we write

Figure 2.26 Inductive step used to construct a space-ﬁlling curve.

2.10 Exercises

55

γ γ

Figure 2.27 Graphs of real and imaginary parts of 0 , 1 . Grey lines included for comparison

purposes.

γn(t) = xn(t) + iyn(t)

so xn , yn : [0, 1] → [0, 1], then Figure 2.27 shows the graphs of x0, y0 and x1 , y1 . The slopes of the parts of these graphs that are not horizontal become increasingly steep as n increases. Nevertheless, the limit is continuous – because of uniform convergence, and more intuitively, because the intervals on which such slopes occur become arbitrarily small (though there are increasingly many such intervals). Exercise 15 shows that the unit hypercube in Rn is also the continuous image of an interval in R. In fact, far more complicated sets can be the images of space-ﬁlling curves. The last word in such results is the Hahn–Mazurkiewicz Theorem: proved by Hans Hahn and Stefan Mazurkiewicz. This states that a set S is the image of a continuous map from [0, 1] to a Hausdorff topological space if and only if S is compact, connected, locally connected, and metric. See Hocking and Young [9], page 129.

2.10

Exercises 1. Let z0 ∈ C and a, b ∈ R be arbitrary. Draw the set consisting of all z ∈ C satisfying the following conditions. In each case, state whether the set is open, closed, or neither. (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x) (xi) (xii)

1 < |z| < 2 re z ≥ 0 and im z ≤ re z re z ≥ 0 and 1 < |z| < 2 re(zz0 ) > 0 a < arg(z − z0 ) < b where −π |z − z¯0 | = |¯z − z0| |z − z¯0 | < |¯z − z0| |z − z¯0 | ≤ |¯z − z0 | re(z2 ) > 0 re(z2 ) ≤ 0 re(z2 ) > 1 re(z2 ) ≤ 1

21 ; i/3, 49(1 + i) Let γ1 = [i, 0] and γ2 = [0, 1]. Show that γ1 + γ2 is deﬁned, but γ2 + γ1 is not. Let γ1 = [0, 1] and γ2 = [1, 0]. Show that γ1 +γ2 and γ2 + γ1 are both deﬁned, but

(i) (ii) (iii) (iv) (v) 7. 8.

they are different paths.

9. Let S be a subset of C. If z, w ∈ S, deﬁne z ∼ w if and only if there is a path from z to w. Show that ∼ is an equivalence relation. The equivalence classes are components of S. If S is open and non-empty, show that each component is a domain.

10. Let S be a path-connected subset of C, and let f : S → C be a continuous function. Prove that f (S) is path-connected (even though it may not be open). 11. Give explicit functions for paths that describe the curves of Figure 2.28 in the direction indicated by the arrows. (All subpaths are parts of circles or line segments; 0 < ε < R and x1 , x2, y1 , y2 are positive reals.) 12. Let S be a subset of C. A point z ∈ C is a boundary point of S if z is a limit point of S and also a limit point of the complement C \ S. The boundary ∂ S of S is the set of all boundary points of S. In the following cases, describe ∂ S and state whether ∂ S is path-connected. Draw a picture in each case. (i) (ii) (iii) (iv) (v) (vi) (vii)

S = {z ∈ C : 1 < |z| < 2} S = {z ∈ C : z ± = 0} S = {z ∈ C : z = x + iy where x, y ∈ Q} S = {z ∈ C : 0 ≤ re z ≤ 1, 0 ≤ z ≤ 1} S = the intersection of the sets S in (iii) and (iv) S = {z ∈ C : z ± = iy where y ∈ R, y ≤ 0} S = the intersection of the sets S in (vi) and (ii)

2.10 Exercises

Figure 2.28 Six paths in

57

C: write down formulas for them.

Figure 2.29 Why can paths like this not be used to construct a space-ﬁlling curve?

13. In the following cases the boundary of S (see Exercise 12) can be described as the image of a path. Draw a picture of S and specify a function γ giving such a path. (i) (ii) (iii) (iv)

S = {z ∈ C : | z| ≤ 1, im z ≥ 0} S = {z ∈ C : 1 ≤ |z| ≤ 2, im z ≥ 0} S = {z ∈ C : 0 ≤ re z ≤ 1, 0 ≤ im z ≤ 1} S = {z ∈ C : 1 ≤ |z| ≤ 2, 0 ≤ im z ≤ re z}

14. In the construction of a space-ﬁlling curve in Section 2.9, why can we not just take γn to be a path that zigzags n times across U2 , as in Figure 2.29? Justify your answer.

15. Let U3 be the unit cube in R3 . Show that there exists a continuous map from [0, 1] onto U3 . (Hint: the hard way is to construct paths with similar properties to those in Section 2.9. The easy way is to observe that U3 = U2 × [0, 1]. Map [0, 1] × [0, 1] onto U2 × [0, 1]; then compose with γ : [0, 1] → [0, 1] × [0, 1].)

58

Topology of the Complex Plane

16. Let Un be the unit hypercube in Rn. Show that there exists a continuous map from [0, 1] onto Un . 17. Let m, n > 0 be integers. Prove that there exists a continuous map from Un, even when m < n.

Um onto

18. If γn is a sequence of step paths in U2, can the union of their tracks equal U (rather than just being dense in U2 )? Prove your answer correct. 19. Does there exist a continuous map γ : [0, 1] Justify your answer.

→ U2 that is one-one as well as onto?

3

Power Series

Many of the more important functions studied in real analysis, such as the exponential and trigonometric functions, are most conveniently deﬁned using power series. The familiar power series for the sine and cosine go back at least to around 1400 with the work of Madhava of Sangamagra, whose discovery of these series was reported in the 1501 Yuktibhasa of Jyesthadeva. This book also included Gregory’s series tan−1 x = x −

x3 3

5

7

+ x5 − x7 + · · ·

for the inverse tangent function: The ﬁrst term is the product of the given sine and radius of the desired arc divided by the cosine of the arc. The succeeding terms are obtained by a process of iteration when the ﬁrst term is repeatedly multiplied by the square of the sine and divided by the square of the cosine. All the terms are then divided by the odd numbers 1, 3, 5, . . . . The arc is obtained by adding and subtracting respectively the terms of odd rank and those of even rank.

Both Madhava and Jyesthadeva were members of the Kerala school of Indian mathematics. The series were rediscovered in the West in 1676, when Newton stated them in a letter to Henry Oldenburg, Secretary of the Royal Society. They were derived again by Abraham de Moivre in 1698, by James Bernoulli in 1702, and were widely used by Euler in the 1730s. Power series are, if anything, even more important when we move from real to com∑ plex functions. In the 1820s Cauchy made considerable use of power series anzn of a complex variable z. In particular, any real function having a power series development automatically gives rise to a complex function with the corresponding power series. This provides a natural method for extending real functions to the complex case. In the 1840s Weierstrass showed how to base the theory of complex functions on power series. In this chapter we develop some elementary properties of sequences and series of complex numbers, mostly by direct analogy with the real case, and then specialise to a deeper study of power series.

3.1

Sequences For our purposes, sequences are required only as a stepping-stone towards series. A (complex) sequence is a function f:

N→C

60

Power Series

where as usual N denotes the natural numbers {0, 1, 2, . . . }. It is convenient to write zn

= f (n)

and to arrange the numbers zn, called the terms of the sequence, in order as z0 , z1 , z2 , . . . , zn , . . . Alternative notation

(zn ) (n ≥ 0)

or just (zn) are often used for brevity, and it is sometimes helpful to start the sequence at 1 instead of 0, like this: z1 , z2 , z3 , . . . , zn , . . .

Now the corresponding f maps N \ {0} to C. D E FI N I T I O N

3.1. A sequence (zn ) tends to the limit z as n tends to ∞ if, given any real

ε > 0, there exists a natural number N( ε) such that n > N(ε) implies |zn − z| < ε A sequence that tends to a limit is convergent.

This deﬁnition is identical to the usual one for real numbers, except that z, zn may be complex, and the absolute value is as deﬁned for C. We write lim zn

n→∞

or

=z

zn → z as n → ∞

The geometric content of this deﬁnition is that for sufﬁciently large n, all terms zn lie inside arbitrarily small discs centred on z, as in Figure 3.1. This again is reminiscent of the real case.

Figure 3.1 A complex sequence converging to a limit.

3.1 Sequences

61

The problem of ﬁnding limits of sequences of complex numbers can be reduced directly to the real case: 3.2. Let (zn ) be a sequence of complex numbers, with zn R). Let z = x + iy (x, y, ∈ R). Then LEMMA

lim zn

n→∞

= xn + iyn , (xn, yn ∈

=z

if and only if

= x and nlim →∞ yn = y Proof. Suppose that limn→∞ zn = z, and let ε > 0. There exists N ∈ N such that |zn − z| < ε if n > N. For any w = u + iv we have ± ± |u| ≤ u2 + v2 = |w| |v| ≤ u2 + v2 = |w | Therefore, taking w = zn − z, for all n > N we have |xn − x| ≤ |zn − z| < ε |yn − y| ≤ |zn − z| < ε so xn → x and yn → y. Conversely, suppose that xn → x and yn → y. Given ε > 0 there exist M, N ∈ N such lim xn

n→∞

that

|xm − x| ≤ ε/2 for all m ≥ M |yn − y| ≤ ε/2 for all n ≥ N

= max(M, N). If m > R then |zn − z| ≤ |xn − x| + |yn − y| ≤ ε/2 + ε/2 = ε so zn → z.

Let R

Since (xn ) and (yn ) are real sequences, their limits x and y are also real and can be found using the techniques of real analysis. Example 3.3. Let zn (n in2 1)−1. Does zn converge? If so, to what? Separate zn into real and imaginary parts, thus:

= +

+

zn = (n + in2 + 1)−1

= (n + 1)[(n + 1)2 + n4 ]−1 − in2 [(n + 1)2 + n4 ]

using (1.8) of Chapter 1. Then xn = yn = so

n+1 (n + 1)2 + n4

−n2

(n + 1)2 + n4

→ 0 as n → ∞ → 0 as n → ∞

zn → 0 + i · 0 = 0

62

Power Series

A similar idea gives the complex version of the General Principle of Convergence: 3.4 (General Principle of Convergence). A sequence (zn) tends to a limit z as n → ∞ if and only if for all ε > 0 there exists N(ε ) ∈ N such that

THEOREM

m, n > N implies |zm

− zn | < ε (3.1) Proof. First, let zn → z. Then there exists N = N(ε) such that | zn − z| < ε/ 2 for all n > N. If m, n > N then |zm − zn | < |zm − z| + |z − zn | < ε/2 + ε/2 = ε so (3.1) holds. Conversely, assume (3.1) holds. Let xn

= re(zn ), yn = im(zn). If m, n > N then |xm − xn| ≤ |zm − zn| < ε |ym − yn| ≤ |zm − zn| < ε

By the General Principle of Convergence for real sequences (that is, the same statement for a real sequence using the real absolute value), it follows that xn → x and yn → y for some x, y ∈ R. By Lemma 3.2, zn → z = x + iy. Example 3.5. Use the General Principle of Convergence to show that the sequence deﬁned by

zn converges. Compute

³n √ ² = i 2 + 3 −6 4i

² ³ ³ ´ ´² ´ 3 − 4i m 3 − 4i n ´´ ´ − 6 ´ |zm − zn| = ´ 6 ´² ³m ´ ´² ³n ´ ´ ´ ´ ´ ≤ ´´ 3 −6 4i ´´ + ´´ 3 −6 4i ´´ ² ³ m ² ³n

=

5 6

≤ 2·

+

² ³r

5 6

5 6

where r = min(m, n). Since 56 < 1 we can make this less than any given ε > 0 by taking r large enough. Note that trying to tackle this question using Lemma 3.2 directly does not work out very easily, although the intrepid reader who converts (3 − 4i)/6 into polar coordinates√stands a better chance). It may of course be shown from the deﬁnition that zn → i 2.

3.2 Series

3.2

63

Series Given a sequence (zn), we can form another sequence of partial sums deﬁned by n µ

sr =

r =0

zr

= z0 + z1 + · · · + zn

If sn tends to a limit s ∈ C, we deﬁne

∞ µ r =0

zr

= s = nlim →∞ sn

and say that the series

∞ µ r =0

zr

converges to s. A series that converges to some s is said to be convergent. We often also write s = z0 + z1 + z2 + · · ·

when convenient, but note that this is a deﬁnition of the + · · · notation, since inﬁnite additions do not have any meaning until we give them one. A series that is not convergent is called divergent. By way of this deﬁnition, any question about the convergence of a series can be turned into one about the corresponding sequence (sn) of partial sums. For example, Theorem 3.4 applied to (sn ) yields:

∑∞

3.6. A series N(ε ) ∈ N such that

r =0 zr

LEMMA

m, n >

∑n

converges if and only if, for all

´ ´ ´ µ ´ ´ n ´ N(ε) implies ´´ zr ´´ < ε ´r =m+1 ´

ε>

0, there exists

= sn − sm . Now use Theorem 3.4. We can replace m + 1 by m here, by working with N + 1 if necessary. This is usually

Proof. Observe that more convenient.

C O RO L L A RY 3.7.

If

r =m +1 zr

∑∞

r =0 zr

converges then zr → 0 as r

Example 3.8. Let

zn

Does ∞ r =0 zr converge? We have

=

²

3 − 4i 6

→ ∞.

³n

´ n ² ´ ³´ n ´² ´µ 3 − 4i ³r ´ µ ´ 3 − 4i r ´ ´ ´ ´ ´ ´ ´≤ ´ ´r =m ´ r =m ´ 6 6 n ² ³r µ

=

r =m

5 6

64

Power Series

This series is a (ﬁnite) geometric series, whose sum is known to be

¶² ³m 5 6

² ³ n+1· ² 5

1−

6

which obviously may be made less than any required ε

³ 5 −1 6

> 0 by taking m, n large enough.

WA R N I N G 3.9. There is an ancient, venerable, and politically incorrect joke about two country yokels who each buy horses and want to be sure which is which. So they cut the tail off one. Next day, someone has cut the tail off the other. So they try again with an ear . . . with the same result. Finally, in desperation, one says to the other ‘Tell you what: you take the black one and I’ll take the white one.’ For some reason, many students become confused between sequences and series. There is no reason to fall into this trap, because:

series are the ones that begin with

∞ µ r =0

This does not eliminate all sources of confusion, but it’s a good start. As with sequences, it is sometimes preferable to start at 1 rather than 0. The series then takes the form

∞ µ r =1

or

zr

z1 + z2 + z3 + · · ·

whose precise deﬁnition is left to the reader. Variations, such as

∞ µ r =3

zr

are also feasible. However, this is no more than ∞ r=0 zr where we set z0 = z1 = z2 To simplify notation, we often abbreviate such sums to

µ

= 0.

zr

when the limits of summation are clear. The summation of complex series may, if desired, be reduced to equivalent real series. Applying Lemma 3.2 to the partial sums, we obtain an immediate proof of: 3.10. Let zr = xr + iyr where xr , yr ∈ ∑ ∑ both real series yr and yr converge. If so, LEMMA

µ

zr

=

µ

R. Then

xr + i

µ

yr

zr converges if and only if

65

3.2 Series

Example 3.11. 11 Let zn We have

= (i)n /n2. Does ∑ zr converge? xn

¸

=

0 (−1)n/2 /n2

if n is odd if n is even

Ignoring the zero terms, xn is an alternating series (its terms are alternately positive and negative) whose terms tend to zero, so it converges. ∑ Similarly, yn = im(zn) yields a convergent alternating series. Hence zr converges.

In the same manner, considering real and imaginary parts, or by mimicking the usual proof for real series, we get: L E M M A 3.12.

Then

Let

ar and

br be convergent series, and let c be a complex number.

µ

(ar + br )

µ

= ar + µ µ ca r =c ar

µ

br

When verifying that a complex series converges without computing its actual sum, it is often better to concentrate on the modulus rather than the real and imaginary parts. By analogy with the real case, we introduce: 3.13. A series ∑ |zr| is convergent. D E FI N I T I O N

THEOREM

zr is absolutely convergent if and only if the series

3.14. An absolutely convergent series is convergent.

Proof. Let zr be absolutely convergent and let zr = xr + iyr . Then |xr | ≤ |zr | and |∑yr | ≤ |zr |.∑But ∑ |zr| is convergent, so by the comparison test for real convergence, |xr| and ∑|yr | are convergent. Since absolute convergence implores convergence in ∑ the real case, xr and yr are convergent. The result follows by Lemma 3.10.

=

/

Example 3.15. zn (in n2) converges because (Compare this proof with the previous example.)

|zn | =

(1/n2 ) converges.

Theorem 3.14 is useful; because it lets us verify convergence of many complex series ∑ zr by reference to the real series | zr |. Since the latter has positive terms there is a complex version of:

3.16 (Comparison Test). Let absolutely convergent. If THEOREM

|br| < K |ar |

for some positive K and integer N, then

ar and (r

br be complex series, with

ar

> N)

br is absolutely convergent, hence convergent.

66

Power Series

There is also a complex version of: THEOREM

3.17 (Ratio Test). Let

that

ar be a complex series with non-zero terms such

|ar | = λ r→∞ | ar − 1| lim

λ < 1, the series ∑ ar is absolutely convergent. If λ > 1, the series is divergent. If λ = 1, it may be convergent or divergent. Proof. For λ < 1, let ρ = 21 (λ + 1). Then λ < ρ < 1 and there exists N such that |ar|/|ar−1 | < ρ (r > N) If

Hence

|ar| < ρ|ar−1 | < ρ2 |ar−2| < · · · < ρr−N |aN | ∑ ∑ and ar converges by comparison with ρ r for real ρ < 1. In the case λ > 1, we see that for some N |ar |/|ar−1 | > 1 (r > N)

(r

> N)

By Corollary 3.7, ar cannot converge because the terms ar do not tend to zero. ∑ ∑ 2 The ﬁnal case λ = 1 applies to 1/r and 1/r . The ﬁrst diverges, the second converges. So both possibilities may occur.

3.3

Power Series We can now give a formal introduction to one of the central ideas in complex analysis: D E FI N I T I O N

3.18. Let z0 ∈ C. A series of the form

∞ µ n=0

an(z − z0) n

(3.2)

with coefﬁcients an ∈ C is a power series about z0 .

By the change of variable z± = z − z0 we can usually reduce properties of power series to the case z0 = 0. Here, convergence is governed by the following results:

3.19. (i) If a power series anzn converges for z = z1 ² = 0, then it converges absolutely for all z with |z| < |z1 |. ∑ (ii) If a power series an zn diverges for z = z2, then it diverges for all z with |z| > |z2| .

THEOREM

Proof. (i) If a power series an zn converges then |anzn | → 0 as n → ∞, by Corollary 3.7. Thus there exists K ∈ R such that |anzn | < K for all n. If | z| < |z1 | then q = | z/z1| < 1. Now

|anzn | = |an zn1||z/z1 |n < Kqn ∑ so by the comparison test, |an zn| converges.

3.3 Power Series

(ii) If |z| > |z2 | and anzn converges then by (i), ∑ n tion. So anz diverges.

67

These results lead to an important concept: D E FI N I T I O N 3.20.

R

Let

= sup{|z| : there exists z such that |an zn| converges}

(allowing R = ∞ if no real supremum exists) then it follows at once that

µ

n

¸

an z

converges for |z| diverges for |z|

(We cannot yet say what happens for | z| convergence of the series, and the set

=

R

R.) We deﬁne R to be the radius of

{z ∈ C : |z| < R} is the disc of convergence, classically often called the circle of convergence because geometrically it is the interior of a circle centred at the origin. In extreme cases this may be just the origin when R = 0, or the whole of C when R = ∞. In the general case (3.2) where z0 is arbitrary, we apply the same deﬁnition with z replaced by z± = z − z0 . Now the conditions

µ

an(z − z0) n

¸

converges for |z − z0| diverges for |z − z0|

R

determine the radius of convergence R, and the disc of convergence is

{z ∈ C : |z − z0| < R} + + +· · · =

||

When |ar |/|ar−1 | tends to a limit as r tends to inﬁnity, the radius of convergence can be computed as follows. Let |ar| = l lim r→∞ | ar −1 | (which is positive or zero). Then

|arzr | = l|z| r →∞ |a r −1zr −1| By the ratio test, for l > 0 the series is convergent for l |z| < 1 and divergent for l| z| > 1. Hence the radius of convergence is 1 /l. Put another way, the radius of convergence of ∑ lim

ar zr is

lim

r →∞

provided this limit exists. If l

|ar−1 | |ar|

= 0 then the radius of convergence is ∞.

68

Power Series

Example 3.22. If an

= 1/n so ∑ an zn = ∑ zn/n, then |ar−1 | = 1 lim r →∞ |a r |

so the radius of convergence is 1. In general, the ratio |ar |/|ar −1| may not tend to a limit. Sometimes it is possible to spot the radius of convergence by native wit, but when all else fails we can use a powerful technique that works in all cases: THEOREM

3.23. The radius of convergence of

an zn is given by

1/R = lim sup |an| 1/n

(3.3)

Proof. Deﬁne R using (3.3). Suppose ﬁrst that |z| < R. Then we can choose ρ such that |z| < ρ < R. By the deﬁnition of lim sup, we have 1/ρ > |an| 1/n for all n larger than some N. Hence | an | < 1/ρ n , so

|an zn| = | an ρn| · |z/ρ|n < |z/ρ|n ∑∞ n n=N | z/ρ| converges. By the comparison

∞ a n zn Now | z/ρ| < 1 so test, n=N ∑∞ converges, so n=0 an zn converges. Next suppose that |z| > R. Choose ρ such that | z| > ρ > R. Then 1/ρ < |an |1/n for ∑ ∑ n all n larger than some N. By comparison with |z/ρ|n, ∞ n=N a n z diverges. Example 3.24. The series

µ

( −1)n

zn n

has radius of convergence R, where

= lim sup(1/n)1/n = 1 so R = 1. The series converges for | z| < 1 and diverges for |z| > 1/R

1. When |z| further analysis is required, for which we do not yet have the technique. ∑ zn The series has radius of convergence R, where n! 1/R

= lim sup(1/ n!) 1/n = 0

Therefore R = ∞ and the series is absolutely convergent for all z ∈ C. The series

µ

similarly have R

(−1)n

z2n 2n!

µ

(−1)n

z2n+1 (2n + 1)!

= ∞, so they are absolutely convergent for all z ∈ C.

=1

69

3.4 Manipulating Power Series

In the next chapter we use the last three power series above to deﬁne the complex exponential, cosine, and sine functions. In order to derive some of their basic properties, we will use a theorem on the multiplication of power series. Roughly, this says that if we multiply them ‘term by term’ and collect terms in the right way, we get a series converging to the product. More precisely: 3.25. Suppose that a, b respectively. Let THEOREM

Then

cr

an and

bn are absolutely convergent, with sums

= a0br + a1br− 1 + · · · + arb0

(3.4)

cn is convergent, and its sum is ab.

Proof. This is a direct extension of the corresponding result in real analysis. For readers unfamiliar with this theorem, we give details in Section 3.5.

3.4

Manipulating Power Series The results derived so far let us calculate with power series ‘as if they are inﬁnite poly∑ n ∑ n nomials’, provided they are absolutely convergent. To see this, let an z and b nz be power series with radii of convergence Ra , R b respectively. Let |z| < min(R a, Rb ). By Lemma 3.12,

µ

By Theorem 3.25,

a n zn +

¹µ

µ

an zn

bn zn =

º ¹µ

b n zn

µ

(an + bn)zn

º

=

µ

c n zn

where cr is deﬁned by (3.4). If we replace the inﬁnite sum by a ﬁnite one, n1 , these formulas become the usual ones for addition and multiplication of polynomials. It is this feature that makes power series so useful: we can calculate with them relatively easily. In fact, we can use familiar algebraic methods. We take advantage of this to exhibit some important features of the complex exponential and trigonometric functions. D E FI N I T I O N 3.26. The complex exponential and trigonometric functions are deﬁned by three power series:

exp z

=

µ zn µ

n!

z2n (2n)! µ z2n+1 sin z = (−1)n (2n + 1)!

cos z

=

(−1)n

These series are motivated by the corresponding real series. They are absolutely convergent for all z ∈ C by Example 3.24.

70

Power Series

We begin with the expression cos θ we obtain cos θ where cr cr Now i2

+ i sin θ of Section 1.7. Adding the power series,

+ i sin θ =

µ

r /2 = (−1)r! (r −1) /2 = i(−1)r!

if r is even if r is odd

= −1, i3 = −i, i4 = 1, so ir = (−1)r /2 ir = i( −1)(r −1)/ 2

Hence

µ

cr θ r

=

cr θ r

if r is even if r is odd

µ ir θ r r!

= exp iθ

We thus have the important formula cos θ + i sin θ

= exp iθ Therefore the polar coordinate form z = r(cos θ + i sin θ ) for a complex number z can be written as z = r · exp iθ , or more brieﬂy as reiθ . (We deﬁne powers za of complex numbers in Section 7.3, and this will justify the notation exp iθ = eiθ .) Next, we apply the formula for the product of two power series to evaluate exp z · exp w where z, w

∈ C. We obtain exp z · exp w

= = =

²µ n ³ ²µ n ³ z w n!

∞ »µ n µ 1

n!

1

zr w n−r

r! (n − r!)

n=0 r =0 ³ ∞ 1 »µ n ² µ n n=0

n!

r =0

r

zr wn−r

By the Binomial Theorem this is

∞ 1 µ n=0

n!

(z + w)n

= exp(z + w)

This proves exp z · exp w = exp(z + w)

¼ ¼

3.5 Products of Series

3.5

71

Products of Series ∑

We sketch a proof of Theorem 3.25 above: if an and ∑ with sums a, b respectively, then ab = cn, where cr

bn are absolutely convergent,

= a0br + a1br− 1 + · · · + arb0

The proof may be easier to follow using Figure 3.2, which represents all possi∑ ble cross-products as bt on a square grid. The idea is that partial sum of cn is the sum of terms in a triangular region like the one shown in light shading. In contrast, ∑ n ∑ n products of partial sums of anz and bn z correspond to terms in a rectangle, such as the one shown in dark shading. Our main task is to estimate the sum of terms in suitable triangular regions by approximation them by rectangles. Here are the details. ∑ ∑ Let | an | = A, |bn| = B. Given ε > 0 we can choose N large enough so that all of the following conditions hold. (To do this, choose an N for each condition and take the maximum of the three Ns chosen.)

´» N ¼ » N ¼ ´ ´ µ ´ µ ε ´ ´ an bn − ab ´ < (i) ´ ´ ´ A+ B+ 1 n=0 n=0

(ii) (iii)

2N µ

n=N +1 2N µ n=N +1

|an | < A + Bε + 1

|bn | < A + Bε + 1

Figure 3.2 Terms of

an zn ,

bn zn, and their product.

72

Power Series

Then

´ ´ ´ 2N ´ 2N » N ¼ » N ¼´ ´» N ¼ » N ¼ ´ ´ ´ µ ´ ´ ´µ µ µ ´ ´ ´ ´ c − ab´ ≤ ´µ c − µ a an bn − ab´ bn ´ + ´ ´ ´ ´ n n n ´ ´ ´ ´ ´ ´ n=0

n=0

n=0

n=0

n=0

n=0

which from Figure 3.2 is less than or equal to

⎛ ⎝

2N µ n=N +1

⎞» ⎞ ¼ ⎛ 2N ¼ » N N µ µ µ |an |⎠ |an | ⎝ |bn|⎠ + |bn | + n=0

n=0

n=N+1

< A +εBB+ 1 + A +εBA+ 1 + A + εB + 1 = ε Therefore

3.6

ε

A+B+1

cn converges to ab.

Exercises 1. Determine whether the following sequences converge, and ﬁnd the limits of those that converge. (i) (ii) (iii) (iv) (v) (vi)

((1 + i)n ) ((1 + i)n /n) ((1 + i)n /n! ) (1/(1 + i) n) (n/(1 + i) n) (n!/(1 + i)n )

(i) (ii) (iii) (iv) (v) (vi)

(zn ) (zn /n) (n!zn ) (zn /n!) (zn /nk ) where k is a positive integer. (a(a − 1) · · · (a − n + 1)zn /n! ) where a is a ﬁxed complex number.

2. For what values of z ∈ C does each of the following sequences converge?

3. Let a ∈ R have decimal expansion

a = a0 .a1 a2 . . . an . . . where each an is an integer and 0 ≤ ai the sequence (an ) converges.

≤ 9 for n ≥ 1. Find all values of a for which

4. Let zn = 21 (in + (−i) n). Write down the ﬁrst few terms of the sequence (zn). Derive similar expressions for the nth terms of the sequences that begin as follows: (i) (ii) (iii) (iv)

1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, . . . 1, 0, 0, 0, −1, 0, 0, 0, 1, 0, 0, 0, −1, 0, 0, 0, 1, 0, − 21 , 0, 13 , 0, − 41 , 0, . . . 1, 0, 1 12 , 0, 32 , 0, 1 41 , 0, 45 , 0, 1 16 , 0, . . .

...

3.6 Exercises

5. Let (un) be a convergent sequence in v±n + v±±n where

v±n

=

73

C. Let vn = ( nr=1 ur)/n. By writing vn = ⎛

1 µ ⎝ ur ⎠ n √

v±±n

r≤ n

=

µ 1 ⎝ ur ⎠ n √ n 0, we ﬁrst choose any r such that |z0 | < r < R. Then convergent and, as in Lemma 3.6, there exists N = N(ε) such that

∞ µ

n=N+ 1

nan rn−1 is

|nan rn−1 | < ε/4

Since |z0 | < r, if z is close enough to z0 to ensure that | z| < r also, then

|±2 | ≤

∞ µ

n=N+1

2n|an| rn−1 < ε/ 2

Furthermore, ±1 is a polynomial in z so ±1 such that

(4.6)

→ 0 as z → z0. We can therefore ﬁnd δ > 0

|z − z0 | < δ implies |±1 | < ε/2

(4.7)

We now choose z close enough to z0 to ensure that both (4.6) and (4.7) hold, and then

´ ´ ´ f (z) − f (z0 ) ´ ´ ´ = |±1 + ±2| ≤ |±1 | + |±2| = ε − g(z ) 0 ´ z − z0 ´ 2

+ ε2 = ε

Therefore f ± (z0 ) = g(z0) as claimed.

Theorem 4.20 is immensely important, for it tells us not only about the ﬁrst derivative of a power series, but about all higher derivatives as well. We just repeat it as often as we want to get: 4.21. All the higher derivatives f ±, f ±± , f ±±± , . . . , f (k), . . . of a power series an zn exists for z within the disc of convergence, and

C O RO L L A RY

f (z) =

f

(k)

(z) =

=

∞ µ n=k

∞ µ n=k

n(n − 1) · · · (n − k + 1)an zn−k n! anzn−k (n − k)!

Proof. Use induction on k.

Replacing z by z − z0 in Corollary 4.21, we ﬁnd that if a power series f (z) = an(z − z0 )n has disc of convergence | z − z0 | < R, then inside this disc of convergence, all the higher derivatives of f exist, and f (k) (z) = Putting z = z0 in this series we get

∞ µ n=k

n!

(n − k)!

an (z − z0 )n−k

f (k) (z0 ) = k!ak which gives yet another important corollary:

87

4.6 A Glimpse Into the Future

C O RO L L A RY 4.22

(Taylor’s Theorem). If f (z) =

an (z − z0 )n for |z − z0|

< R, then

(k)

= f k(z! 0)

ak

We can thus express f (z) as a convergent Taylor series f (z) =

∞ f (n) (z ) µ 0 n=0

Example 4.23. Suppose that f (z) Differentiating, we ﬁnd that

f ± (z) = f ±± (z) =

n!

(|z − z0|

zn

= 1/(1 − z) = 1 + z + z2 + · · ·

1 (1 − z)2 2 (1 − z)3

for | z|

0)

88

Differentiation

Trivially,

φ ±(x) =

±

0 2x

(x ≤ 0) (x > 0)

and an easy calculation gives

φ(x) − φ(0) = 0 φ ±(0) = xlim →0 x ± Thus φ exists, and is even continuous at 0. But φ±± (0) does not exist because φ± (x) − φ± (0) = ± 0 (x ≤ 0) 2 (x > 0) x More generally, the function

φ (x) =

±

0 xn+ 1

(x ≤ 0) (x > 0)

is differentiable n times everywhere, but not n + 1 times at the origin. We shall see later that there is no similar method for piecing together complex functions to obtain this type of behaviour. For a real function, there are only two ways to approach a limit point x0 : from the left (smaller values of x) and from the right (larger values of x). We can piece real functions together quite happily, provided we make sure that the left and right derivatives are the same to whatever extent we deem necessary.

4.6.2

Bad Behaviour of Real Taylor Series Indeed, we can go further. Above, we pieced together 0 on the left and xn+ 1 on the right to obtain a function that can be differentiated n times but not n + 1 times. But we can even ﬁnd functions like ± (4.8) φ(x) = 0e−1/x (x(x ≤> 0)0) which has all derivatives in agreement at the origin, when approached either from left or right. This ‘Frankenstein’s monster’ creation is patched up well, but there is something unnatural about it. Because all derivatives are zero at the origin, its Taylor series about the origin is

∞ φ(n)(0) µ n=0

n!

= 0 + 0x + 0x2 + · · · + 0xn + · · · = 0

This converges for all real x, but its sum is not equal to φ(x). Thus in the real case we can ﬁnd inﬁnitely differentiable functions that are not equal to their Taylor series. Despite our reference to Frankenstein, there is nothing mysterious about this. It just means that the remainder term Rn (x) in the Taylor series

φ(x) = a0 + a1x + a2 x2 + · · · + anxn + Rn(x) does not tend to zero as n tends to ∞. Indeed, here R n(x) = φ(x) for all n. (In the

theory of differentiable manifolds, the ability to patch real functions together smoothly like this is actually very useful. It provides topological ﬂexibility. In this context, the complex case really is more complicated.)

4.6 A Glimpse Into the Future

4.6.3

89

The Blancmange function Real analysis has even more grey areas, inhabited by functions that are continuous everywhere yet differentiable nowhere. Bernard Bolzano found such a function around 1831, but it was not published until 1922. Charles Cellérier discovered one around 1860, which was published in 1890, shortly after his death. The ﬁrst such function to be published was constructed by Weierstrass in 1872, as the sum f (x) =

∞ µ

n=0

an cos(bn π x)

where 0 < a < 1, b is a positive odd integer, and ab > 1 + 3π/2. The smallest value of b satisfying these conditions is b = 7. Weierstrass’s function is continuous by uniform continuity of the terms in the sum. Term-by-term differentiation leads to a series that does not converge, and Weierstrass gave a rigorous proof that the function is nowhere differentiable. Later Godfrey Harold Hardy improved the conditions to 0 < a < 1, ab > 1. Cellérier’s example is very similar to Weierstrass’s. Figure 4.2 shows how irregular this function is. We describe a simpler example that goes by the name of the blancmange function. Let G(x) be the distance from x ∈ R to the nearest integer. This has a graph like the teeth of a saw: ± x (0 ≤ x ≤ 21 ) G(x) = 1 − x ( 12 ≤ x ≤ 1)

and G is periodic with G(x + n) = G(x) for any integer n. It is not differentiable at x = 21 n for any integer n. Moreover, |G(x) ≤ 12 | for all x ∈ R. It follows that the function Gn (x) = ( 41 )nG(4nx) is not differentiable at x = 21 ( 41 )m for any integer m, and satisﬁes 0 ≤ Gn(x) ≤ 12 ( 14 )n . The sum b(x) =

∞ µ n=0

Gn (x)

is sketched in Figure 4.3, which shows why it is called the blancmange function.

Figure 4.2 Weierstrass’s continuous nowhere differentiable function.

90

Differentiation

Figure 4.3 Top left: Successive halvings of the sawtooth function. Bottom left: Adding the sawtooth functions gives successive stages of the blancmange function. Right: An advanced stage of the construction. We show only the unit interval; the full function is periodic with period 1 and deﬁned on the whole of R.

Intuitively, the curve is a fractal, a set with detailed structure on all scales of magniﬁcation, Mandelbrot [13] and Falconer [4]. This is clear because the graph of the blancmange function has smaller blancmanges everywhere. Some are clearly recognisable, while others are disguised by being sheared vertically as they stand on straight line segments that lie at an angle. In general, there are two kinds of points on the graph: those at dyadic rationals x = k2−n for integers k, n with n ≥ 0, where the graph comes down vertically from the left and goes up vertically to the right. Here, under high magniﬁcation the graph looks like a half-line pointing upwards. Everywhere else, the graph is locally a tiny blancmange, sheared on a small line segment. This is easy to see when the segment is not too steep, but not when it is nearly vertical. A rigorous proof requires extra effort, and we sketch one now: P RO P O S I T I O N

4.24. The blancmange function is continuous everywhere and differen-

tiable nowhere. Proof. The estimate 0 ≤ b(x)

∞ µ 1 n=0

1 n 2 ( 4)

1

= 1−2 1 = 23 4

holds, so by uniform continuity, as in Section 2.9 on space-ﬁlling curves, b is continuous everywhere. To prove it is differentiable nowhere, observe that in order for b to be differentiable at α , the expression γ (x) = (b(x) − b(α))/(x − α ) must tend to a limit. We construct a sequence (αn ) → α such that γ (αn) does not converge. Under each line segment of Gn there are two identical teeth of Gn+1 of length ( 41 )n , so we can ﬁnd αn = α ³ ( 14 )n such that Gm (αn ) = Gm (α)

(m ≥ n + 1)

The gradient of the straight bit of the tooth of Gn and larger teeth is Gm (αn ) − Gm (α )

αn − α

=1

(m ≤ n)

4.6 A Glimpse Into the Future

Hence

γ (αn ) = b( ααn ) −− αb(α) ≤ n

91

∞ G (α ) − G (α ) µ m n m n=0

αn − α

is a sum of n + 1 terms, each ³1. Now γ (αn ) is an odd integer when n is even and an even integer when n is odd. Therefore (γ ( αn )) cannot tend to a limit. Having created the bad function b, its antiderivative b1(x)

=

¶ 0

x

b(t)dt

is differentiable once everywhere (with derivative b) but is twice differentiable nowhere. Repeating this process inductively we obtain a function bn that is differentiable n times everywhere (with derivative bn−1 ) but is (n + 1) times differentiable nowhere. Looking back at the idea of a space-ﬁlling curve, we begin to see why our intuitive ideas of continuity and differentiability need modifying to ﬁt with the formal epsilondelta deﬁnitions of continuity and limits. The space-ﬁlling curve is constructed as a formal limit of curves made up of straight line segments, and in the construction we take successively smaller line segments that change direction. In the case of the blancmange function, successive curves change direction at dyadic rationals of the form k2−n where k, n are integers and n ≥ 0. The limit function is continuous everywhere and differentiable nowhere, with the slope at the dyadic rationals being vertically down to the left and vertically up to the right. The space-ﬁlling path constructed here is the limit of graphs whose real and imaginary parts also turn suddenly at dyadic rationals (Figure 2.26) yet the distance travelled in each step gets smaller and smaller (Figure 2.27). This is how it manages to go through every point in the unit square. Our strategy to deal with this problem is simple. When we calculate integrals in complex analysis, we use ‘contours’ made up of a ﬁnite number of ‘smooth paths’, which have non-zero continuous derivatives to ensure that the formal approach resonates with our intuitive idea of tracing the contour in real time using a ﬁnger.

4.6.4

Complex Analysis is Better Behaved Real analysis is a very hairy subject indeed. But what is the relevance of such bizarre functions in complex analysis? The answer is: NONE WHATSOEVER. They have been mentioned only to contrast the real case with the complex case. There do, however, exist very simple complex functions that are continuous but differentiable nowhere. An example is f (z) = |z| , whose continuity is easily proved, but fails to satisfy the Cauchy–Riemann Equations at any point. There are also functions, such as f (z) = |z| 2, that are differentiable only at isolated points (here, the origin), and are thus not differentiable twice at those points. But that is the end of the line. If a complex function is differentiable once in a domain, then (as we prove in Chapter 10) it is differentiable any number of times, has a convergent Taylor series, and is equal to its Taylor series. The reason for this much nicer behaviour is that in computing the derivative

92

Differentiation

f ± (z) = lim

z →z0

f (z) − f (z0 ) z − z0

z can approach z0 from any direction in the complex plane. The existence of the limit is therefore a much stronger condition than it is in the real case, and precludes patching together different functions at z0. The key to the whole theory of complex analysis is the fact that every differentiable complex function has a power series expansion near any point of its domain, so is equal to its own Taylor series as in Corollary 4.22. This will be established by a roundabout route in Chapter 10, but it is worth waiting for, and it underlines our emphasis on power series. They are not just good examples of differentiable functions: in a very genuine sense they are the only examples.

4.7

Exercises 1. From ﬁrst principles, differentiate: (i) f (z) = z2 + 2z (ii) f (z) = 1/z (z ² = 0) (iii) f (z) = z3 + z2

2. Show that f (z) = |z| is continuous everywhere and differentiable nowhere. Show that f (z) = | z|2 is continuous everywhere and differentiable at the origin but nowhere else. 3. Differentiate: (i) 3z2 + 2z3 (ii) (z2 + 3iz − 4)(z3 − 7i) (iii) (z5 − 13iz2)99 (iv) (z3 + 3)5 (z4 + (26 − 11i)z)12 (v) (z2 + 3)/(4z3 + 5 − 3i)2

²

(vi) 4. Let

z +n z −n

Show that

³n

, n ∈ N, z ² = n

·

fn (z) = 1 +

z ¸n n

²

(n − 1)z f ± (z) = fn−1 n

What do you notice as n → ∞ ?

³

n

5. Let Cπ = {z ∈ C : z ² = x ∈ R, x ≤ 0 be the ‘cut plane’ with the negative real axis removed. Prove that Cπ is a domain. Deﬁne r : Cπ → C by (r(z))2 = z and re r(z) > 0. Prove that r is continuous on Cπ , and hence show from ﬁrst principles that r ± (z) = 1/(2r(z)). 6. Let f (z) be a polynomial in z ∈ C. Prove that the function g(z) = f (z¯) is differentiable everywhere, but that h(z) = f (z) is differentiable at 0 if and only if f ± (0) = 0.

4.7 Exercises

93

7. In each of the following cases, for f deﬁned on the domain D, ﬁnd explicit formulas for u(x, y), v(x, y) where f (z) = u(x, y) + iv(x, y), where z = x + iy and all of x, y, u, v are real. (i) f (z) = 1/z, D = {z ∈ C : z ² = 0} (ii) f (z) = |z| , D = C (iii) f (z) = z¯, D = C Show that u, v satisfy the Cauchy–Riemann Equations everywhere in (i) and nowhere in (ii), (iii). 8. Verify the Cauchy–Riemann Equations for the functions u(x, y), v(x, y) deﬁned in the given domains by (i) u(x, y) = x3 − 3xy2 , v(x, y) = 3x2y − y3 (x, y ∈ R) (ii) u(x, y) = sin x cosh y, v(x, y) = cos x sinh y (x, y ∈ R) (iii) u(x, y) = x/(x2 + y2), v(x, y) = −y/ (x2 + y2¹ ), (x2 + y2 ² = 0) (iv) u(x, y) = 21 log(x2 + y2 ), v(x, y) = sin−1 ( y/ x2 + y2) (x > 0) In each case, state a complex function whose real and imaginary parts are u(x, y) and v(x, y).

9. For z = x + iy, let

f (z) =

f (0) =

x3 (1 + i) − y3(1 − i) x2 + y2 0

(z ² = 0)

Show that f is continuous at the origin, the Cauchy–Riemann Equations are satisﬁed there, yet f ± (0) does not exist. Why does this not contradict Theorem 4.12?

10. Let f (z) = |xy| for z = x + iy. Show that the Cauchy–Riemann Equations are satisﬁed at the origin yet f ± (0) does not exist. Why does this not contradict Theorem 4.12? 11. Consider

xy2 (x + iy) x2 + y4 f (0) = 0 f (z) =

(z = x + iy ² = 0)

Verify that limz→z0 ( f (z) − z0 )/(z − z0) = 0 as z → 0 along any straight line, z = (a + ib)t, t ∈ R. This does not prove that f ±(0) = 0, however. By considering z → 0 along the path z(t) = t2 + it, show that f is not differentiable at 0. (This shows that when computing f ± it is not enough to consider a limit taken along certain speciﬁc paths or types of path. An entire neighbourhood of the point concerned must be considered.)

12. For each of the following, compute f ± (γ (t)), γ ± (t), ( f γ )± (t) and verify directly that ( f γ ) ±(t) = f (γ ± (t)) γ ± (t)

(i) f (z) = z2 , γ (t) = t3 + it4 (z ∈ C, t ∈ [0, 1]) (ii) f (z) = 1/z, γ (t) = cos t + i sin t (z ² = 0, t ∈ [0, 2π ]) (iii) f (z) = 1 + z + z2 + · · · , γ (t) = t + it2 ( |z| < 1, t ∈ [0, 21 ])

94

Differentiation

∑ n 13. Suppose that f (z) = anz is convergent for all z ∈ C and satisﬁes f ± f (0) = 1. Find an for all n ≥ 0. Consider the derivative of g where

=f

and

g(z) = f (c − z)f (z)

for c ∈ C and deduce that

f (a + b) = f (a) + f (b)

for all a, b ∈ C. Compute f (1) to ﬁve decimal places. (A calculator or computer may be used, but electronic assistance is not necessary.)

14. Show that for all α

∈ C the power series ∞ α (α − 1) · · · (α − n + 1) µ fα (z) = zn n! n=1

converges for |z| < 1, and that in this domain its derivative is αfα (z)/(1 + z). Is the radius of convergence 1 for all α ∈ C? By differentiating fα (z)fβ (z) /fα+β (z) or otherwise, show that Deduce that for | z|

fα+β (z) = f α(z)f β (z)

< 1,

fn (z) = (1 + z)n

for every integer n (positive or negative), and that ( f1/n (z))n

= (1 + z)

for all positive integers n. (This power series can be used to deﬁne (1 + z)α for domain | z| < 1.)

α ∈ C and any z in the ∑

15. Assume that the two power series s(z) = anzn and c(z) = bn zn are convergent for all z ∈ C, and that they satisfy the relations s± (z) = c(z), c± (z) = −s(z). Deduce the identities

= −an−2 /(n(n − 1)) bn = −an−2 /(n(n − 1)) If further s(0) = 0, c(0) = 1, determine s(z) and c(z) completely. By differentiation, an

or otherwise, prove that

(c(z))2 + (s(z))2

=1

16. For a positive integer n the Bessel function of order n is deﬁned by Jn(z)

=

∞ (−1)r ( 1 z)n+2r µ 2 r= 0

r !(n + r) !

Show that this converges for all complex z and satisﬁes the differential equation z2

d 2y dz2

+ z dy + (z2 − n2 )y = 0 dz

95

4.7 Exercises

Verify the following: (i) Jn−1 (z) + Jn+1(z) (ii) (iii) (iv) (v) (vi)

= 2nz Jn(z)

Jn± (z) = nz Jn (z) − Jn+1 (z) Jn± (z) = 21 (Jn−1 (z) − Jn+1(z)) Jn± (z) = Jn−1(z) − zn Jn (z) d n n dz (z Jn(z)) = z Jn−1(z) J2 (z) − J0 (z) = 2J0±± (z)

17. Deﬁne f : R

→ R by

f (x) =

±

0 e−1/x

(x ≤ 0) (x > 0)

Show that f is differentiable arbitrarily many times, and that f (n) (0) n ∈ N.

=

0 for all

5

The Exponential Function

An abstract theory of functions may be intellectually diverting, but it pays its way by its more concrete applications by deﬁning speciﬁc functions of particular interest and establishing their properties. As a step in that direction we now discuss the complex versions of the usual (real) exponential and trigonometric functions exp, sin, and cos, deﬁning them as power series. We also introduce complex versions of the usual related functions such as tan and cosec. From these deﬁnitions we develop some basic properties of these functions. Euler’s famous formula exp(iθ )

= cos θ + i sin θ

follows at once from these deﬁnitions, and shows that the fundamental function here is exp. All of the others may be deﬁned very simply in terms of it. We also discuss complex generalisations of the hyperbolic functions, such as sinh, cosh, tanh, which again have simple deﬁnitions in terms of exp. Most of the material generalises directly from the real case, and will be presented in a compact form. In addition to deriving the standard formulas, we place some emphasis on convincing the reader that the power series do indeed represent the usual real functions, and that in particular the geometric interpretations of sin and cos in trigonometry are valid. We also derive further properties of these functions that are peculiar to the complex case, including their relation to hyperbolic functions. The complex version of the logarithm requires deeper analysis, and is held over to Chapter 7.

5.1

The Exponential Function Deﬁnition 3.26 deﬁnes the exponential function for complex numbers as the power series exp z =

∞ zn ± n=0

n!

(5.1)

which is absolutely convergent for all z ∈ C. We may therefore differentiate term by term. The result is the same series, proving that d exp z = exp z dz

(5.2)

5.1 The Exponential Function

97

With a little ingenuity we can use (5.2) to prove the formula exp(z1 + z2) = exp(z1 ) exp(z2 )

(5.3)

derived in a cumbersome way in Chapter 3. Consider f (z) = exp(z) exp(c − z) for c ∈ C. Differentiate: f ± (z)

= exp± (z) exp(c − z) + exp(z) exp± (c − z) = exp(z) exp(c − z) + exp(z) exp(c − z)(−1) =0 By Theorem 4.14 f is constant; the constant must equal f (0) = exp(c). So exp(z) exp(c − z) = exp(c) Put z = z1 , c = z1 + z2 to obtain (5.3).

We wish to use the customary notation ez for exp z. To avoid ambiguities we must show that √ when z is rational, say z = m/n, this agrees with the usual real exponential em/ n = n em . We do this as follows. First, observe that exp(x) > 0 for all x ∈ R. This is obvious from the power series when x ≥ 0. But when x < 0 we know that exp(−x) exp(x) = exp(0) = 1, so exp( −x) = 1/ exp(x) > 0. Therefore the nth root of exp(x) has an unambiguous meaning for any positive integer n. Next, deﬁne the real number e = exp(1) = 2.718 281 . . .

(5.4)

by an easy computation using (5.1) with z = 1. Using (5.3) and induction on n we obtain exp(nz) = (exp(z)) n for any positive integer n. Therefore exp(n) = (exp(1))n = en Clearly

Then

so that

exp(0) = 1 exp(n) exp(−n) = exp(n − n) = exp(0) = 1 exp(−n) = (exp(n))−1

= (en)−1 = e−n

Now, for any rational m/ n, (n > 0) we have (exp(m/n))n

= exp(nm/n) = exp(m) = em

98

The Exponential Function

so that

exp(m/n) = (em )1/n

= em/n

Thus the notation ez = exp(z) does not conﬂict with the standard notation for powers of e, so we may (and do) use it from now on. Equation (5.3) becomes

Putting z = x + iy this gives

ez1 ez 2

= ez +z

ex+iy

= exeiy

1

(5.5)

2

Here ex is the usual real exponential. So we know how ez behaves provided we also understand eiy . We study both of these in the next two sections.

5.2

Real Exponentials and Logarithms We brieﬂy recall some standard properties of ex when x is real. Clearly e x > 1 + x for x > 0, which implies that ex > 0 for x ≥ 0, and ex → ∞ as x → ∞. Also e−x = 1/ex for x < 0, so ex > 0 for all x ∈ R and ex → 0 as x → −∞. The derivative of ex is ex > 0 so ex is monotonic increasing for all x. That is, e x deﬁnes a continuous strictly increasing function from R onto R+ = {x ∈ R : x > 0}. By the Intermediate Value Theorem ex has a continuous strictly increasing inverse function, deﬁned to be the natural logarithm log : R+

→R

with the property y = log x

⇐⇒ x = ey

From (5.5) we obtain

= log x1 + log x2 (x1, x2 > 0) Let y = log x, y0 = log x0 (x, x0 ∈ R+ ). Since log is continuous, log x − log x0 y − y0 1 1 lim = lim y = = y y x→x y→y e − e x − x0 e x0 log(x1 x2)

0

0

Therefore d 1 log x = dx x

0

0

(5.6)

5.3 Trigonometric Functions

5.3

99

Trigonometric Functions Deﬁnition 3.26 also deﬁnes the complex sine and cosine functions by the power series cos z = sin z =

∞ ± n=0

∞ ± n=0

(−1)n

z2n (2n)!

(5.7)

(−1)n

z2n+1 (2n + 1)!

(5.8)

We know these are absolutely convergent for all z ∈ C. Putting −z for z in (5.7, 5.8) we see that cos is an even function and sin is an odd function. That is, cos(−z) = cos(z)

sin(−z) = − sin(z)

Also cos(0) = 1 sin(0) = 0

Differentiating term by term, d cos z = sin z dz d sin z = − cos z dz

(5.9) (5.10)

Term-by-term addition, as in Chapter 3, leads to Euler’s formula eiz = cos z + i sin z

(5.11)

= einz for any integer n, equation (5.11) implies De Moivre’s Formula (cos z + i sin z)n = cos nz + i sin nz (5.12) This may be used to obtain rapid derivations of formulas for cos nθ and sin nθ when θ ∈ R, by equating real and imaginary parts of both sides (Exercise 6).

Since (e iz )n

Euler’s Formula has the famous corollary ei π

= −1

(5.13)

Formulas (5.11) and (5.13) seem surprising when we ﬁrst encounter them, because there is no obvious connection between e and π . Historically, these special numbers arose in very different areas of mathematics – natural logarithms and the circumference of a circle. Moreover, it is hard to see why i, which arises from yet a third area, polynomial equations, should link them together. But in fact, there is a simple reason why these results hold, based on yet another area of mathematics: differential equations, applied to uniform rotation in the plane. We discuss this in Section 5.6. If anything, this dynamic

100

The Exponential Function

explanation adds to the beauty of these formulas by showing that they are at the core of a remarkable uniﬁcation of distinct mathematical concepts. Replacing z by −z in (5.11) gives e−iz = cos(−z) + i sin(−z)

= cos z − i sin z

(5.14)

From (5.11) and (5.14), cos z = 21 (eiz + e−iz ) sin z =

1 iz (e 2i

(5.15)

− e−iz)

(5.16)

From (5.15, 5.16), and (5.5), we obtain the usual addition formulas for sin and cos, now valid for all complex numbers z1, z2 : sin(z1 + z2 ) = sin z1 cos z2 + cos z1 sin z2

sin(z1 − z2 ) = sin z1 cos z2 − cos z1 sin z2

cos(z1 + z2 ) = cos z1 cos z2 − sin z1 sin z2

cos(z1 − z2 ) = cos z1 cos z2 + sin z1 sin z2

Putting z1

(5.18) (5.19) (5.20)

= z2 = z in (5.20) gives

cos2 z + sin2 z = 1

5.4

(5.17)

(5.21)

An Analytic Deﬁnition of π Historically the real number π was deﬁned as the ratio of the circumference of a circle to its diameter, and only later did its importance for trigonometric functions emerge. We reverse the process, deﬁning π analytically and eventually showing in Section 7.1 that our deﬁnition agrees with the geometric one. The idea is to deﬁne π/2 as the ﬁrst positive real solution of the equation cos x = 0. The problem is to show that there is such a thing. We know that cos and sin are continuous functions. Also, cos(2) = 1 −

22 4!

4n−2

6

4n

2 + 26! − · · · − (4n2 − 2)! + (4n) ! − ··· 4n−2

= 1 − 2 + 32 − · · · − 2(4n)! [4n(4n − 1) − 4] − · · · < 1 − 2 + 32 < 0

But cos(0) = 1. By the Intermediate Value Theorem, cos t0 = 0 for some t0 Let k be the greatest lower bound of {t ∈ R : t > 0, cos t = 0}. By continuity, cos k = 0

By the deﬁnition of k, if 0 ≤ x < k then cos x > 0. We deﬁne

π = 2k

∈ (0, 2).

5.5 The Behaviour of Real Trigonometric Functions

101

Then the number π has been uniquely deﬁned by the properties π/2 > 0, cos(π/2) = 0 and 0 ≤ x < π/2 implies cos x > 0. Since cos(2) < 0, it follows that 0 < π < 4. This is a crude estimate, and we improve it in Exercises 17 and 18 below.

5.5

The Behaviour of Real Trigonometric Functions

π/2. Since dxd sin x = cos x, it follows that

We know that cos x is positive for 0 ≤ x < sin is strictly increasing on [0, π/2]. Since

π + cos2 π = 1 2 2 and cos π/2 = 0, we must have sin π/2 = ²1. But since sin is increasing on [0, π/2], we must have π sin = 1 sin2

2

Now (5.18) implies that sin

²π

³

− x = cos x 2

(5.22)

Hence cos decreases monotonically from 1 to 0 in [0, π/2]. Using (5.17) and (5.19) repeatedly, we can deduce the behaviour of sin and cos in the intervals [π/2, π ], [π , 3π/2], and [3π/2, 2π ]. We obtain

²π

³

+ x = − sin x ³ ² π2 + x = cos x sin 2 sin(π + x) = − sin x

cos

and so on. We tabulate the results in Table 5.1, where a ³ b means ‘strictly increasing from a to b’, and a ´ b means ‘strictly decreasing from a to b’. From the table, cos 2 π

=1 sin 2π = 0

Therefore cos(x + 2π ) = cos x cos 2π

− sin x sin 2π = cos x sin(x + 2π ) = sin x cos 2π + cos x sin 2π = sin x

(5.23) (5.24)

leading to cos(x + 2nπ ) = cos x sin(x + 2nπ ) = sin x

for all n ∈ Z. So sin and cos are periodic with period 2 π . In particular the behaviour in the table repeats on each interval [2nπ , (2n + 2)π ].

102

The Exponential Function

Table 5.1 Behaviour of sin and cos in intervals of length

interval

cos

[0, π/ 2] [ π/2, π ] [ π , 3 π/2] [3π/ 2, 2π ]

1´ 0 0 ´ −1 −1 ³ 0 0³ 1

π/2.

sin

³1 ´0 ´ −1 −1 ³ 0

0 1 0

In this way, purely formal considerations show that sin x and cos x have their usual geometric properties for all real x, at least in outline. The ﬁnal remaining step is to identify the point cos θ + i sin θ = eiθ on the unit circle |z| = 1 with the point θ radians from 1 in an anticlockwise direction. We indicate two different approaches in Section 5.6 and Exercise 8 in Chapter 6. We also discuss the topic in detail in Chapter 7. More precise computations of the values of these functions may be performed to any desired degree of accuracy. By inspection, it may be seen that for real x (positive or negative) the terms of the power series for sin x and cos x always alternate in sign. From the standard theory for alternating real series, the sum of the ﬁrst n terms alternately overestimates and underestimates the actual limit. This lets us make very precise estimates of the trigonometric functions. Example 5.1. Compute ei to four decimal places. We have e i cos(1) i sin(1). Now

=

+

cos(1) Considering partial sums:

= 1 − 21! + 41! − 61! + 81! − · · ·

cos(1) < 1 1 cos(1) > 1 − = 0.5 2! 1 1 cos(1) < 1 − + = 0.541 66 . . . 2! 4! 1 1 1 cos(1) > 1 − + − = 0.540 27 . . . 2! 4! 6! 1 + 1 − 1 + 1 = 0.540 30 . . . cos(1) < 1 − 2! 4! 6! 8! Therefore Similarly

cos(1) = 0.5403 (to 4 decimal places) sin(1) = 0.8415 (to 4 decimal places)

Therefore ei

= 0.5403 + 0.8415i (to 4 decimal places)

103

5.6 Dynamic Explanation of Euler’s Formula

5.6

Dynamic Explanation of Euler’s Formula Euler’s formula(s) linking e, i, π seem surprising, but the existence of such a link, and even the details of how it goes, can be deduced very naturally from standard ideas in the theory of differential equations. This section is intended as motivation, and to help explain why such a connection exists. It can be made rigorous by setting up the necessary ideas formally. It is not used later in the book and can be skipped. Consider the linear differential equation dz dt

= iz

(z ∈ C)

(5.25)

on C. Equation (5.2) and the chain rule show that a solution z(t) with initial condition z(0) = 1 is z(t) = eit

This solution is unique since the difference w between any two solutions satisﬁes dw dt

=0

so w(t) is constant, and must equal w(0) = 0. Geometrically, (5.25) corresponds to a point particle z(t) moving in the plane C, and it states that the velocity vector z± (t) is at right angles to the position z(t), and the speed is | z(t)| , Figure 5.1. (Here ± indicate the time-derivative. The right angle arises from the factor i in the equation.) That is: the particle always moves at right angles to the line joining it to the origin. Intuitively, the particle must move along the unit circle. To verify this, we prove that z± (t) is always tangent to the unit circle S , or equivalently that z(t) ∈ S for all t. The key point is that | z(t)| 2 is conserved; that is, it is constant. Compute: d |z(t)|2 dt

= dtd z(t)z¯(t) = z± (t)¯z(t) + z(t)z¯± (t) = (ieit )(e−it ) + (eit )(−ie−it ) = i − i = 0

so | z(t)| 2 is constant. Since it equals 1 at time 0, we know that |z(t)|2 Therefore z(t) always lies on the unit circle.

Figure 5.1 Uniform motion round the unit circle.

=

1 for all t.

104

The Exponential Function

The speed of motion is

|z± (t)| =

´

z± (t)z¯± (t) =

µ

(ieit )(−ie−it ) = 1

so the particle moves with unit speed (anticlockwise). After time t, the particle therefore moves a distance t along the unit circle in the anticlockwise direction; that is, through an angle of t radians. At time t = π is has gone a distance π round the unit circle – that is, halfway round. So it must lie at the point diametrically opposite the starting point z(0) = 1. This point is −1. So z(π ) = −1; that is, eiπ = −1, the miraculous formula. More generally, at time t its position on the unit circle makes angle t radians with the real axis at the origin, so its Cartesian coordinates are (cos t, sin t). Interpreted as a complex number, this point is cos t + i sin t. Therefore eit

= cos t + i sin t

and we recover Euler’s formula (5.11). This approach is closely related to Section 7.1.

5.7

Complex Exponential and Trigonometric Functions are Periodic We are used to the real sine and cosine functions being periodic, with period 2π , and we have proved that the same property holds for their complex generalisations. Euler’s formula shows that their periodicity is inherited by the complex exponential function, as we now describe. D E FI N I T I O N

5.2. A complex function f : S → C has period ρ

∈ C if

f (z + ρ )

= f (z) (z, z + ρ ∈ S) Obviously, if ρ is a period of f , so is nρ for any positive integer n such that mρ ∈ S for m ≤ n, and with similar conditions we can also take n negative. For the complex exponential, S = C, and e2π i = cos 2π + i sin 2π = 1 so

= ez e2π i = ez · 1 = ez Therefore 2π i is a period for exp. So is 2nπ i for any n ∈ Z. There are no others: P RO P O S I T I O N 5.3. The complex number ρ is a period of exp if and only if ρ = 2n π i for some n ∈ Z . Proof. If ρ is a period then eρ = ez +ρ /ez = 1. If ρ − u + iv then 1 = eu+iv = eueiv = eu (cos v + i sin v) Taking the modulus,

ez +2π i

105

5.8 Other Trigonometric Functions

1 = | eu|| cos v + i sin v| = eu

By properties of the real exponential proved in Section 5.2, u = 0. So now cos v + i sin v = 1

Therefore cos v = 1, sin v = 0. By Table 5.1, v = 2nπ , (n ∈ Z). We also have: P RO P O S I T I O N 5.4.

The exponential function ez is non-zero for any z

Proof. For any z ∈ C, ez e−z

= e0 = 1.

∈ C.

We now investigate sin and cos for periodicity in C. Certainly 2nπ is a period of both, because the proofs of (5.23) and (5.24) for real x also work for complex z. 5.5. The complex number = 2nπ for some n ∈ Z.

P RO P O S I T I O N

ρ

ρ is a period of sin or cos if and only if

Proof. Since sin(z + π/2) = cos z by (5.17), it follows that ρ is a period for cos if and only if it is a period for sin. Then cos(z + ρ )

= cos z sin(z + ρ ) = sin z

for all z ∈ C. But then ei(z +ρ )

= cos(z + ρ) + i sin(z + ρ) = cos z + i sin z = eiz so iρ is a period of exp. By Proposition 5.3, iρ = 2nπ i for some n ∈ Z, so ρ = 2nπ . We can also ﬁnd the zeros of sin and cos: P RO P O S I T I O N 5.6.

where n ∈ Z.

∈ C. Then cos z = 0 if and only if z = (n + 21 )π sin z = 0 if and only if z = nπ

Let z

Proof. Since sin(z + π/2) = cos z by (5.17), the second equation implies the ﬁrst. Now sin z = 0 if and only if (eiz − e−iz )/2i = 0. This holds if and only if eiz − e−iz = 0, that is, if and only if e 2iz = 1. So 2iz = 2nπ i. Therefore z = nπ as claimed.

5.8

Other Trigonometric Functions If z µ = (n + 12 )π then cos z µ = 0, so we may deﬁne tan z If S = { z ∈ C : z µ = (n + 12 )π , n ∈ differentiable function. Its derivative is

sin z = cos z

Z } then S is a domain, and tan

(5.26) : S

→ C is a

106

The Exponential Function

d cos z cos z dzd sin z − sin z dz d tan z = dz cos2 z cos z · cos z − sin z · (− sin z) = cos2 z 2 z + sin2 z = cos cos 2z 2 = 1 + tan z

Similarly we deﬁne cos z (z µ = nπ ) (5.27) sin z 1 sec z = (5.28) (z µ = (n + 21 )π ) cos z 1 cosecz = (z µ = nπ ) (often also written csc z) (5.29) sin z These are all differentiable functions (on the stated domains) whose derivatives may again be calculated in the usual manner. All of the standard formulas relating these trigonometric functions may be deduced from properties of sin and cos, hence also apply for complex values of the variables. For example, using (5.17) and (5.19), we obtain cot z =

tan(z1 + z2 ) =

tan z1 + tan z2 1 − tan z1 tan z2

provided that z1 , z2 , z1 + z2 ∈ S. This implies that tan(z + π ) = tan z, so π is a period for tan. It is easy to see that the only periods of tan are nπ for n ∈ Z. The reader is encouraged to develop all of his or her favourite trigonometric formulas for the complex case, including the basic properties of cot, sec, and cosec.

5.9

Hyperbolic Functions As in the real case, we deﬁne sinh z = 21 (ez − e−z )

for z ∈ C. Differentiating,

cosh z = 12 (ez + e−z )

d sinh z = cosh z dz d cosh z = sinh z dz Properties of the hyperbolic functions, analogous to those of trigonometric functions (such as addition formulas for sinh(z1 + z2 )) follow either by direct computation, or by using the obvious identities sin iz = i sinh z

cos iz = cosh z

5.10 Exercises

107

For example, cosh2 z − sinh 2 z = cos2 iz − (−i sin 2 iz)

= cos2 iz + i sin2 iz =1

The functions tanh, coth, sech, and cosech (or csch) are deﬁned in the obvious way. We leave it to the reader to discover their properties, including zeros and periods. Hyperbolic functions occur in expressions for the real and imaginary parts of sin z and cos z. Thus let z = x + iy. Then sin z = sin(x + iy)

= sin x cos iy + cos x sin iy = sin x cosh y + i cos x sinh y

(5.30)

cos z = cos x cosh y − i sin x sinh y

(5.31)

Similarly

5.10

Exercises 1. Express the following in the form a + ib for real a, b: (i) exp(i) (ii) e2+iπ (iii) 1/ exp(2 + iπ )

2. Express the following in the form a + ib for real a, b: (i) sin(i) (ii) cos(i) (iii) sinh(i) (iv) cosh(i) (v) cos(π/4 − i) (vi) tan(1 + i) 3. Differentiate the functions deﬁned as follows: (i) exp(z2 + 2z) (ii) 1/ exp(z) (iii) exp(z2 )/ exp(z + 1) 4. Differentiate the functions deﬁned as follows: (i) tan(z2) (ii) sinh(z + 2) exp(z3 ) (iii) sin(z) cosh(z) exp(z)

5. Use the identity eiθ eiφ = ei(θ +φ) to derive the usual formulas for cos( θ sin(θ + φ ). By a similar method, show that 1/(cos θ

+ i sin θ ) = cos θ − i sin θ

+ φ ) and

108

The Exponential Function

(

)3

6. Use the identity eiθ = e3iθ to give the usual formulas for cos(3θ ) and sin(3θ ). Derive similar formulas for cos(4θ ), sin(4θ ), cos(5 θ ), and sin(5θ ). 7. Draw the following paths: (i) γ (t) = e−it (t ∈ [0, π ]) (ii) γ (t) = 1 + i + 2e −it (t ∈ [0, 2π ]) (iii) γ (t) = z0 + re−it (t ∈ [0, 2π ]), where z0 (iv) γ (t) = t + i cosh t (t ∈ [−1, 1]) (v) γ (t) = cosh t + i sinh t

∈ C and r > 0

8. ‘Osborne’s rule’ states that any formula involving sin and cos has an analogous formula involving sinh and cosh, which is the same in every way except that the product of two sines must be replaced by minus the product of two hyperbolic sines. For each of the following formulas, write down the corresponding formula using Osborne’s rule and verify it from ﬁrst principles: (i) sin 2A + cos A = 1 (ii) sin(A − B) = sin A cos B − cos A sin B (iii) cos(A + B) = cos A cos B − sin A sin B Comment on Osborne’s rule in the light of the formulas cos iz = cosh z

sin iz = i sinh z

9. Show that the complex conjugate of cos z is cos ¯z and that of sin z is sin z¯. Verify the identities

10.

| sin z|2 = 21 (cosh 2y − cos 2x) = sinh2 y + sin2 x = cosh2 y − cos2 x | cos z|2 = 21 (cosh 2y + cos 2x) = sinh2 y + cos2 x = cosh2 y − sin2 x Show that | cos z|2 + | sin z|2 = 1 if and only if z is real, and that cos z is unbounded on C (that is, no K > 0 exists such that | cos z| ≤ K for all z ∈ C). This contrasts with the bound | cos x| ≤ 1 for all real x.

11. Derive formulas for the real and imaginary parts of the following functions of z, and check directly that they satisfy the Cauchy–Riemann Equations: (i) exp z (ii) sin z (iii) cos z (iv) sinh z (v) cosh z 12. Derive formulas for the real and imaginary parts of the following functions of z, specifying the largest domain on which they are deﬁned, and check directly that they satisfy the Cauchy–Riemann Equations: (i) tan z (ii) tanh z (iii) cosec z (iv) cosech z (v) cot z (vi) coth z

5.10 Exercises

109

13. Write tanh(x + iy) in real and imaginary parts and show that tanh(x + iy) is real if and only if y = nπ/2 for n ∈ Z. 14. For each of the functions exp, cos, sin, tan, cosh, sinh, tanh, ﬁnd the set of points on which it assumes: (i) real values; and (ii) purely imaginary values. 15. By considering the real and imaginary parts of 1 + z + z2 + · · · + zn

n+1

= 1 1−−z z

ﬁnd explicit formulas for the sums: (i) 1 + cos x + cos 2x + · · · + cos nx (ii) sin x + sin 2x + · · · + sin nx By similar methods, ﬁnd: (iii) cos x + cos 3x + · · · + cos(2n − 1)x (iv) sin x + sin 3x + · · · + sin(2n − 1)x (v) sin x − sin 2x + · · · + (−1)n sin nx (vi) cos θ + cos( θ + φ) + · · · + cos( θ + nφ ) (vii) sin θ + sin(θ + φ ) + · · · + sin(θ + nφ ) 16. If z(t)

= x(t) + iy(t) is a solution of the differential equation d2 z dt2

+ λz = k0 eiωt (λ, k0, ω ∈ R)

(5.32)

show that x = x(t) = re z(t) is a solution of d 2x dt2

+ λx = k0 cos ωt

(5.33)

By ﬁnding solutions of (5.32) of the form z(t) = keiωt , ﬁnd a solution of (5.33). If k0 is complex, say k0 = k1eiε (k, ε ∈ R), and λ, ω are real, write down the real part of (5.32) to obtain d 2z dt 2

+ λz = k0 cos(ω t + ε)

Show that there is a solution of (5.34) of the form x(t) =

k1

λ − ω 2 cos(ωt + ε )

17. By using the sum of the geometric progression 1 + z + z2 + · · · + zn

= 1 1−−z z

n+1

ﬁnd A n(x) such that 1

1 + x2

= 1 − x2 + x4 − · · · + (−1)nx2n + An (x)

(5.34)

110

The Exponential Function

Verify that the derivative of tan−1 x is 1/ (1 + x2) where tan−1 : R → (−π/2, π/2) is the inverse function of tan : (−π/2, π/2) → R. Hence deduce that tan−1 t

=

0

t

dx 1 + x2

= t − t3 /3 + t5 /5 − · · · + (−1)nt 2n+1 /(2n + 1) +

0

t

A n(x)dx

By estimating the size of the second integral, show that the power series t − t3 /3 + t 5/5 − · · · + (−1)nt 2n+1 /(2n + 1) + · · ·

converges to tan t for |t | < 1. Deduce Gregory’s series

π = 1 − 1 + 1 − 1 + ··· 4 3 5 7

18. Gregory’s series converges very slowly. Classically, better methods for calculating π were obtained using the related series

π = tan−1 1 + tan−1 1 4

2

3

π = 4 tan−1 1 + tan−1

1 4 5 239 Verify these formulas using the addition formula for tan(θ + φ). Use the second to 1 calculate π correct to 5 decimal places. (This requires only one term of tan− 1 239 and ﬁve terms of tan−1 15 .) Series that converge far more rapidly to π are now known. You can ﬁnd them easily on the Internet.

6

Integration

The next part of the grand plan is to deﬁne complex integration by analogy with the real case, and establish the inverse relation between differentiation and integration. Consider a complex function f : D → C on a domain D, and let z0 , z1 ∈ D. The ±b ±z real integral a f (t)dt does not generalise immediately to the complex integral z01 f (z)dz because this expression does not specify how z goes from z0 to z1. To do that, ± we must specify a path γ between them. A sensible notation for the integral is then γ f (z)dz, or ± γ f for short.

±b

In real analysis the value of a deﬁnite integral a f (t)dt depends only on the limits a, b. Complex analysis is more complicated, and the integral along a path may depend on the path as well as on its end points. We give a simple example in Section 6.10 as soon as we have introduced the necessary± concepts and techniques. There are two approaches to deﬁning γ f . The ﬁrst is to build up the theory of complex Riemann sums by mimicking the real case. We do this in Sections 6.1 and 6.2. This approach applies to any continuous function, provided it is integrated along a continuous path γ : [a, b] → C. This is one reason why we built the condition of continuity into the deﬁnition of a path in Chapter 2, by requiring γ to be a continuous map. In the special case where the function γ is continuously differentiable, with derivative γ ± , we derive an explicit formula for the integral:

²

γ

f

=

²

b

a

( f γ (t))γ ±(t)dt

(6.1)

Here z0 = γ (a), z1 = γ (b). This formula makes sense only when γ ± is deﬁned and the expression concerned is Riemann integrable; the requirement that γ should be continuously differentiable is the simplest way to ensure this. To simplify terminology, such a path is said to be smooth. A key result, the Estimation Lemma 6.41, bounds the absolute value of such an integral by the product of the supremum of | f (z)| on γ and the length of the path γ . It turns out that ‘length’ need not be a meaningful concept for a continuous path – another somewhat counterintuitive fact – but it is well-behaved for smooth paths. In Section 6.3 we deﬁne ‘length’ and show that for any smooth path it is given by the simple formula L= Notice that the derivative of for assuming smoothness.

γ

²

a

b

|γ ± (t)|dt

(6.2)

features explicitly in this formula, which is one reason

112

Integration

These equations lead to a second approach, in which the reader is assumed to be familiar with real integration. Why not restrict the theory to smooth paths (or, more generally, piecewise smooth paths – that is, ﬁnite sums of smooth ± paths – which pose no new problems)? Then we can use (6.1) and (6.2) to deﬁne γ f and L. Admittedly, the integrand ( f γ (t))γ ± (t) in (6.1) is complex, but we can reduce the formula to real integrals by writing it as U(t) + iV(t) and deﬁning

²

γ

f

²

=

b

a

U(t)dt + i

²

a

b

V (t)dt

We adopt this method in Section 6.8 as an alternative route to complex integration, allowing Sections 6.1–6.3 to be omitted. This short cut has a price. It means that a couple of proofs later in the chapter must be given in a more technical and less intuitive manner. But this price is not very great, and it leaves the reader with a genuine choice: work through the theory of complex Riemann integration to experience the full analogy with the real case, using continuous paths, or bypass the next three sections and start at Section 6.4, restricting to smooth paths.

6.1

The Real Case For the reader who has chosen to build up the analogy between the real and complex integrals, we begin by recalling the real case. ±b The Riemann integral a φ(t)dt of a real function φ : [a, b] → R is deﬁned in stages. First, subdivide the interval [a, b] to obtain a partition P of [a, b] given by a = t0 < t1 < · · · < tn = b, and choose intermediate points sr in each subinterval t r−1 ≤ sr ≤ t r. Then form the Riemann sum S(P, φ ) =

n ³ r =1

φ (sr )(tr − tr−1 )

The points t 0, t 1, . . . , tn are called the division points of P. Another partition Q is said to be ﬁner than P if the division points of P are all included in those of Q. The following result is well known from real analysis; we quote it without proof. 6.1. Let φ : [a, b] → R be continuous. Then there exists a real number A such that for any ε > 0 there is a partition P ε of [a, b] such that, for every partition P ﬁner than Pε , we have

LEMMA

|S(P, φ) − A| < ε ±b This real number A is denoted by a φ(t)dt, and is the Riemann integral of φ from a

to b. ± The actual computation of ab φ (t)dt is usually performed by antidifferentation:

6.2 (Fundamental Theorem of Calculus). (i) If a real function φ is continuous on [a, b] and F ± = φ , then

THEOREM

²

a

b

φ(t)dt = F(b) − F(a)

113

6.2 Complex Integration Along a Smooth Path

(ii) If φ is continuous on [a, b] and I(x) then I ±

=

= φ.

²

x

a

φ (t)dt

(x ∈ [a, b])

±b

φ

Example 6.3. To compute a t5 dt we do not need to calculate the sums S(P, ) where (t) t5 . Since F(t) t6 6 satisﬁes F ± , Theorem 6.2(i) immediately gives

φ =

= /

²

b

a

t 5dt

= 61 b6 − 61 a6

±

More generally, we can compute ab φ(t)dt by seeking an antiderivative F and appealing to Theorem 6.2(i). Before passing to the complex ± b case, it is helpful to consider a slight generalisation: the Riemann–Stieltjes integral a φ(t)dθ , where θ is a second real function. Here, given a partition P of [a, b] as above, we form the sum S(P, φ, θ ) =

n ³ r =1

φ (sr )(θ (tr ) − θ (tr−1))

From real analysis we have a generalisation of Lemma 6.1: L E M M A 6.4. Let φ , θ be real functions deﬁned on [a, b], such that φ is continuous and ± θ has continuous derivative θ ± . Then the real number B = ab φ(t)θ ± (t)dt satisﬁes the

following condition: Given any ε > 0 there is a partition Pε of [a, b] such that, for every partition P ﬁner than Pε , we have

|S(P, φ, θ ) − B| < ε ±b The limit B of the sum S(P, φ , θ ) is also denoted by a φ(t)dθ . It is the Riemann– Stieltjes integral of φ from a to b with respect ± to θ . Lemma 6.4 tells us that when θ is differentiable, the Riemann–Stieltjes integral ab φ (t)dθ is equal to the Riemann integral ±b ± ± a φ (t)θ (t)dt. The conditions of the lemma ensure that the integrand φ(t)θ (t) exists and is continuous. The Riemann and Riemann–Stieltjes integrals exist under far more general conditions on φ, θ than those we have mentioned, and the reduction from the Riemann–Stieltjes integral to the Riemann integral is then not always valid. The conditions in Lemma 6.4 are all that we require in this book.

6.2

Complex Integration Along a Smooth Path Riemann sums deﬁne complex integrals for continuous functions and paths, but we can derive a simple formula for the integral, which often lets us compute it, if we work with a more special class of paths: those with continuous derivatives.

114

Integration

6.5. A path throughout all of [a, b].

D E FI N I T I O N

γ

: [a, b]

→ C is smooth if γ ± exists and is continuous

This means that if γ (t) = x(t) + iy(t), where x and y are real, then x± and y± exist and are continuous on the whole of [a, b], including one-sided derivatives at the end points. The deﬁnition is intended to formalise the physical sense of drawing a smooth path with a ﬁnger from the starting point to the ﬁnal point. Invoking dynamical imagery, the path γ describes a point particle moving in the plane, whose position at any time t ∈ [a, b] is γ (t). Smoothness ensures that the point γ (t) moves with a speciﬁc velocity at all times t, namely γ ± (t), and that this velocity changes continuously. Our intuition here is not just geometric: it is dynamic. Recall that velocity is usually described as ‘speed and direction’, where speed is the magnitude of the velocity and direction is determined by the time-derivative of position. The speed is |γ ± (t)|, a non-negative real number. When γ ±(t) ² = 0, the coordinates (x± (t), y±(t)) of |γ ± (t)| determine a non-zero vector in the plane, which does indeed point in a speciﬁc direction. What is often ignored is the caveat that when γ ±(t) = 0 this becomes the zero vector, which does not point in a speciﬁc direction. This distinction is important both geometrically and dynamically, and we discuss it further in Section 6.7. We now consider integration of a complex function along a smooth path in C. In this case, the Riemann integral of a complex function f can be approached by analogy ∑ with the limit of the sum φ(sr )(tr − tr−1 ). We simply consider ∑ f (ζr)(zr − zr−1 ), where ζ r , zr are complex, and select ζ r and zr along a path γ in the domain of f as in Figure 6.1. For this purpose we assume: (i) f : D → C is continuous, where D is a domain; and (ii) γ : [a, b] → D is smooth. For any partition P of [a, b], given by a = t0 points tr −1 ≤ sr ≤ tr , form the sum S(P, f , γ ) =

Figure 6.1 A partition of a path in

C.

n ³ r =1

< t1 < · · · < tn = b, and intermediate

f (sr )( γ (tr ) − γ (t r−1 ))

6.2 Complex Integration Along a Smooth Path

115

= γ (tr ), ζr = γ (sr), this Riemann–Stieltjes sum becomes

Writing zr

S(P, f , γ )

n ³

=

r=1

f (ζ r )(zr − zr −1 )

which exhibits the direct analogy with the real case. As in the real case, we rarely calculate an integral using this summation process. One method is to reduce the calculation to two real integrals by taking real and imaginary parts. If ψ : [a, b] → R, let

ψ (t) = U(t) + iV(t)

and deﬁne

²

γ

²

ψ (t)dt =

b

a

(t

∈ [a, b])

U(t)dt + i

²

b

a

V (t)dt

With this convention we derive a complex version of Lemma 6.4: T H E O R E M 6.6. For a continuous complex function f deﬁned on a domain D, and a smooth path γ : [a, b] → D,

²

f

γ

=

²

b a

f (γ ((t))γ ± (t)dt

(6.3)

Proof. Let γ (t) = x(t) + iy(t) (t ∈ [a, b]) and f (z) = u(x, y) + iv(x, y) (z = x + iy ∈ D). Denoting u(x(t), y(t)), v(x(t), y(t)), x± (t), y± (t) by u, v, x±, y± for short, our convention for complex integrals gives

²

a

b

f (γ ((t))γ ± (t)dt =

= =

²

a

²

a

²

b

(u + iv)(x± + iy± )dt

b

(ux ± − vy ± ) + i(vx±

b

ux± dt −

a

²

a

b

+ uy± )dt

vy ±dt + i

²

If we write f (γ (sr )) = ur + ivr and γ (t r ) = xr + iyr , then S(P, f , γ ) =

=

n ³ r =1 n ³ r =1

+i

b

a

vx ± dt + i

² a

b

uy ±dt

(ur + ivr )[(xr + iyr ) − (xr −1 + iyr −1 )] ur (xr − xr −1 ) − n ³ r =1

n ³ r =1

vr (yr − yr−1 )

vr (xr − xr −1) + i

n ³ r =1

ur (yr − yr− 1)

We now match the four integrals and four sums in pairs, and use Lemma 6.4. For instance, n ³ r =1

ur (xr − xr −1)

=

n ³ r =1

u(x(sr ), y(sr ))(x(tr ) − x(tr −1 ))

116

Integration

where both φ (t) = u(x(t), y(t)) and x± (t) are continuous on [a, b]. So given ε > 0, we can ﬁnd a partition P1(ε ) such that for any partition P that is ﬁner than P1 (ε) we have

´ ´ n ² b ´ ε ´³ ´ ´ ux± dt´ < ´ ur (xr − xr −1) − ´ 4 ´ a r =1

Similarly we ﬁnd partitions P 2(ε ), P3 (ε), P4 (ε) such that all ﬁner partitions P give similar inequalities between corresponding pairs of integrals and sums. Taking Pε to have division points all those of P1 (ε), P2 (ε), P 3(ε ), P4 (ε), then whenever P is ﬁner than Pε all four inequalities hold. Therefore

´ ´ ² ´ ´ ´S(P, f , γ ) − b f (γ ((t))γ ± (t)dt´ ´ ´ ´ ´ a

Hence

²

γ Example 6.7. f (z)

f

=

² a

b

< 4ε + 4ε + 4ε + 4ε = ε

f (γ ((t)) γ ± (t)dt

= z2, γ (t) = t 2 + it (t ∈ [0, 1]). See Figure 6.2. ²

γ

f

= = =

²

1

f (γ (t)) γ ±(t)dt

1

(t2 + it)2 (2t + i)dt

1

(t4 + 2it 3 − t2 )(2t + i)dt

0

²

0

²

0

²

1

²

1

= (2t − 4t )dt + i (5t 4 − t 2)dt 0 0 4 1 5 1 3 1 1 6 = [ 3 t − t ]0 + i[t − 3 t ]0 = − 23 + 32 i

Figure 6.2 Path for Example 6.7.

5

3

6.3 The Length of a Path

6.3

117

The Length of a Path The length of a path is deﬁned in terms of approximating polygons, as follows. 6.8. Given an arbitrary path γ : [a, b] → C, take a partition P of [a, b] given by a = t0 < t1 < · · · < t n = b, and calculate the length D E FI N I T I O N

L(γP ) =

n ³ r =1

of the approximating polygonal curve ure 6.3.

|γ (tr) − γ (tr−1 )|

γP

with vertices

γ (t0 ), γ (t 1), . . . , γ (t n),

Fig-

Figure 6.3 Length of a path as the supremum of the lengths of approximating polygons.

The length L(γ ) of γ is the supremum (least upper bound) of the lengths L(γP ) over all such approximating polygons. The length of an arbitrary path need not be ﬁnite:

γ =

∈ + /

Example 6.9. Suppose (t) (t [0, 1]) is deﬁned as follows. Let mn be the midpoint of [1 (n 1), 1 n], so mn (2n 1) (2n(n 1)). Introduce a function where the graph 1 1 of y (t) consists of two straight line segments from ( n+ 1 , 0) to (mn, n ) and then to

/ + / =λ

( n1 , 0). Deﬁne

γ (t) =

µ

+

t + iλ(t) (0 < t ≤ 1) 0 (t = 0)

The image of γ is drawn in Figure 6.4.

Figure 6.4 A path with inﬁnite length.

λ

118

Integration

1 to t = n1 exceeds 2n . Let P be the partition The length of this path from t = n+ 1 1 1 1 0 < n+1 < mn < n < · · · < 2 < m1 < 1. Then the length of the polygonal path exceeds ¶ · 2 2 2 1 1 1 + + ··· + 1 = 2 1 + 2 + 3 + ··· + n n n−1

The latter is twice the harmonic series, so it increases without limit as n increases. Therefore L(γ ) is inﬁnite. In this example, the path is not smooth. The next example shows that even if the path is ‘nearly smooth’, its length can still be inﬁnite. Example 6.10. Consider the path

γ (t) =

µ

t + it sin(π/t) 0

(0 < t ≤ 1) (t = 0)

as in Figure 6.5.

Figure 6.5 A nearly smooth path with inﬁnite length.

A calculation shows that although

γ ±(t) = 1 + i(sin(π/t) + t cos(π/t))(−π/t 2) is continuous for 0 < t ≤ 1, the limit of γ (t) − γ (0) = 1 + i sin(π/t) t

as t tends down to 0 does not exist. Hence γ ± is not continuous on the closed interval [0, 1], so is not smooth on [0, 1] (though it is smooth on any subinterval [k, 1] with 0 < k ≤ 1). We claim this path has inﬁnite length. Let P n be the partition 0 < 1/ n < 1/(n − 21 ) < 1/(n − 1) < · · · < 1/2 < 1/(2 − 21 ) < 1

6.3 The Length of a Path

Since

γ and

γ

¸

¶ · 1 n

¹

1

n−

1 2

119

= 1n + i sinnnπ = 1/n + i sin(n − 12 )π 1 n− 2 n− 2 n−1 = 1 1 + i (−1) 1 n− 2 n− 2 1

1

=

the distance from γ (1/n) to γ (1/(n − 21 )) exceeds 1/ (n − 21 ), which exceeds 1/ n. We need only compute the lengths of alternate segments of the polygonal approximation γn to γ given by Pn to ﬁnd that L(γn) >

+ n −1 1 + · · · + 1

1 n

which again increases without limit.

6.3.1

Integral Formula for the Length of Smooth Paths and Contours When a path is smooth, its length is ﬁnite, and can be calculated by an integral: P RO P O S I T I O N 6.11.

The length of a smooth path γ : [a, b] L(γ )

²

=

a

b

|γ ± (t)|dt

→ C is (6.4)

and this is ﬁnite. We prove this result below, but ﬁrst we note that the integrand |γ ± (t)| is continuous on [a, b], so the real integral L=

²

b

a

|γ ± (t)|dt

certainly exists, and is ﬁnite. We must show that L is the supremum of the lengths L(γP ) of approximating polygons γP. Now L can be closely approximated by sums S(P, φ ) = where P is the partition a = t0

n ³ r =1

|γ ± (sr )|(t r − tr−1)

< t1 < · · · < tn = b, tr−1 < sr < tr , and φ (t) = |γ ± (t)|.

120

Integration

The proof of Proposition 6.11 is greatly facilitated by: 6.12. With the preceding notation, given any ε > 0, there exists a partition Qε such that for any partition P ﬁner than Q ε , the length L(γP ) of the approximating polygon corresponding to P satisﬁes

LEMMA

|S(P, φ) − L(γP )| < ε Proof. By deﬁnition, n ³ S(P, φ) = |γ ± (sr )|(tr − tr−1) r =1

L(γP)

=

n ³ r =1

|γ (t r) − γ (tr−1 )|

Writing γ (t) = x(t) + iy(t), the Mean Value Theorem in real analysis gives x(tr ) − x(tr −1 ) = x± (σr )(tr − tr −1 ) for some σr y(tr ) − y(tr −1 ) = y± (τr )(tr − t r−1 ) for some τr

so

∈ (tr−1 , tr ) ∈ (tr−1 , tr )

γ (t r) − γ (tr−1 ) = (x± (σr) + iy± (τr))(t r − tr−1)

Now x±, y± are continuous on [a, b], and from real analysis they are uniformly continuous since [a, b] is a closed interval. This means that for any ε > 0 there exists δ > 0 such that

∈ [a, b] and |s − t| < δ implies |x± (s) − x±(t)| < ε/2(b − a) and |y± (s) − y± (t)| < ε/2(b − a) (where the same δ works for all s, t). s, t

If we choose Qε to be any partition such that each subinterval [t r−1 , t r ] has length less than δ , then any partition P ﬁner than Qε has the same property. Because sr , σr , τr ∈ [tr −1 , tr ], they are all within distance δ of each other, so

| |γ ± (sr )|(tr − tr−1) − |γ (t r) − γ (tr−1) | | = | |γ ±(sr )| − |x± (σr ) + iy± (τr )| | (tr − t r−1 ) ≤ |(x± (sr) + iy± (sr )) − (x± (σr ) + iy± (τr ))| (tr − tr−1) ≤ (|x± (sr) − x± (σr )| + |y± (sr) − y±( τr )|)(tr − tr−1) < ( 2(bε− a + 2(bε− a )(t r − tr−1) = ε (tr − tr−1 )/(b − a)

where in the third line we use property (1.12) of the modulus. Thus

|S(P, φ ) − L(γP )| ≤
0, has =

² 2π 0

|ireit | =

² 2π 0

rdt = 2π r

122

Integration

6.4

If You Took the Short Cut . . . Some readers will have read the previous three sections, and some will not. For the latter, we now give some basic deﬁnitions that are motivated by the previous three sections: 6.15. A path γ : [a, b] → C is smooth if γ ± exists and is continuous on the closed interval [a, b]. If D is a domain, the integral of a continuous function f : D → C along the path γ is

D E FI N I T I O N

²

f

γ

The length of γ is

=

L(γ )

²

b

a

=

f ( γ (t))γ ± (t)dt

² a

b

(6.10)

|γ ± (t)|dt

(6.11)

A comment on (6.10) and (6.11) is in order to explain what they mean. The integrands f (γ (t))γ ±(t) and |γ ±(t)| are both continuous. The latter is real; the former may be written ± in real and imaginary parts as f (γ (t)) γ ± (t) = U(t) + iV (t) and the integral γ f can then be calculated using two real integrals:

²

γ

f

=

²

b

a

U(t)dt + i

²

a

b

V (t)dt

Since all integrands involved are continuous, the real integrals all exist. At this stage, readers who made either choice are now equipped to proceed to the rest of the book. If you took the short cut, take a quick look at Examples 6.13 and 6.14 to convince yourself that the formula deﬁning length makes sense for lines and circles.

6.5

Further Properties of Lengths Recall from Deﬁnition 2.27 that paths can be added together if their start and end points match up correctly. Using approximating polygons, it is easy to prove that the length function L is additive: P RO P O S I T I O N

6.16. If γ

= γ1 + · · · + γn and L(γr ) exists for 1 ≤ r ≤ n, then L( γ ) = L(γ1 ) + · · · + L(γn )

It is important to understand that the length of a path need not be the same as the length of its image curve. In particular, this happens whenever the path ‘doubles back on itself’, as in the following example.

σ − ρ = −

ρ − → − γ = σ ◦ρ

Example 6.17. Let : [ 2, 2] C be any smooth path. Let : [ 2, 2] [ 2, 2] 3 be given by (t) t 3t, which is a smooth map, and let . Then it can be shown, using (6.4), that L( ) 3L( ). The geometric signiﬁcance of this result is straightforward: as t increases from 2 to 2, the cubic t3 3t increases from 2 to 2,

γ =

σ

6.5 Further Properties of Lengths

123

then decreases from 2 to −2, and ﬁnally increases from −2 to 2 again. So the path traces the curve three times. Next, we consider how a change of parameter affects the length of a smooth path. D E FI N I T I O N 6.18. Smooth paths γ : [a, b] → C, λ : [c, d] → C with the same image curve C are smoothly equivalent parametrisations of C if there is a smooth function ρ : [a, b] → [c, d] with non-zero derivative ρ± (t) ²= 0 for t ∈ [a, b], where ρ(a) = c, ρ(b) = d, and γ = λ ◦ ρ.

This relationship is an equivalence relation, because ρ − 1 exists, is smooth, and also has non-zero derivative, by the inverse function theorem. A smoothly equivalent parametrisation can be imagined as tracing the same curve in the same direction but at different speeds.

P RO P O S I T I O N 6.19. If two smooth paths γ : [a, b] → C, λ : [c, d] → C are smoothly equivalent parametrisations of the same curve C, then they have the same length: L(γ ) = L(λ).

Proof. We have γ = λ ◦ ρ where ρ : [a, b] → [c, d] is a strictly increasing real function with continuous derivative ρ ± on [a, b]. Therefore ρ ± (t) ≥ 0, so ρ ± (t) = |ρ ± (t)| . Let s = ρ (t). Then as t increases from a to b, s increases from c to d and we can substitute ds = ρ ± (t)dt in the integral. The length of the path γ is therefore L(γ ) =

= = = = 6.5.1

²

a

²

a

²

a

²

a

²

a

b

|γ ± (t)|dt

b

|(λ ◦ ρ )± (t)|dt

b

|(λ± (ρ(t))||ρ± (t)|dt

b

|(λ± (ρ(t))|ρ± (t)dt

b

|λ± (s)|ds = L(λ)

Lengths of More General Paths You may be wondering why we require γ to be smooth in the deﬁnition of length in Deﬁnition 6.15? Proposition 6.16 applies to contours made from several smooth pieces, which need not ﬁt together smoothly where they join, so paths that are not smooth can have meaningful lengths. Obviously, Deﬁnition 6.15 cannot be used for continuous paths, because it involves the derivative of γ . But the deﬁnition ‘supremum of lengths of all approximating polygons’, which occurs in the alternative treatment of Section 6.3, makes sense for any path, without assuming smoothness. However, this deﬁnition has its own awkward features. Sometimes it goes wrong in the mild sense that the entire path has inﬁnite length, as in Examples 6.9 and 6.10. However, it gets much worse. Using the

124

Integration

‘approximating polygon’ deﬁnition for length, there are continuous paths γ : [a, b] → C with the disturbing property that the length of any segment of the path, between points c < d in [a, b], is inﬁnite. In fact, we have already met just such a path: the graph of the blancmange function, Figure 4.3. Using dyadic rationals, it is straightforward to construct a sequence of partitions of [c, d] such that the corresponding polygons have lengths tending to inﬁnity. In fact, it is enough to do this for the graph of the blancmange function on the interval [0, 1], because we have already pointed out that this graph contains arbitrarily small copies of the graph of the blancmange function, and a small number times inﬁnity is still inﬁnity. These copies are usually distorted by an afﬁne transformation, but such a distortion keeps the length inﬁnite. Similar remarks apply to space-ﬁlling curves, and to standard fractals such as the snowﬂake curve, Mandelbrot [13], and Falconer [4]. In other words, a path that is continuous but not smooth may not have a meaningful length. This is probably counterintuitive, because we have been trained from an early age to assume that every linear object does have a length. The ancient Greeks worried about the length of the circumference of a circle without ever deﬁning what they were worrying about. It seemed obvious to them that any arc of a circle must have a length; their main aim was to ﬁnd out what that length is. To do so, they tacitly assumed several plausible properties, such as ﬁnding the length by using polygons to approximate the circumference. Eventually mathematicians tightened up the logic, leading to Deﬁnition 6.8 above, and understood what can go wrong. To complete the story, smoothness is not actually necessary for a path or curve to have a well-deﬁned length. There is a more general notion of a rectiﬁable curve, for which the ‘approximating polygon’ deﬁnition is entirely satisfactory. However, the smooth case is all we need in this book.

6.6

Regular Paths and Curves Just as we distinguish between a path γ and its image curve, we must distinguish between the derivative γ ± (t) and a tangent line to the curve. The derivative can be interpreted as the velocity vector at time t for a point γ (t) moving along the curve. If γ ± (t) ² = 0 it deﬁnes a tangent direction, hence a tangent line to the curve. When γ ± (t) = 0 it does not deﬁnes a tangent direction, so the curve may not have a tangent line. Section 6.7 shows some of the things that can then happen. First, some standard terminology: 6.20. Let γ : [a, b] → C be a smooth path. ∈ [a, b] and γ ± (t 0) ²= 0, then t0 is a regular point of γ . ∈ [a, b] and γ ± (t 0) = 0, then t0 is a singular point of γ .

D E FI N I T I O N

If t0 If t0

When the image curve has a well-deﬁned tangent line, it looks smooth: see Proposition 6.22 below. The above discussion leads naturally to a special type of path or curve that will be useful as we proceed, to relate the abstract theory to geometric intuition:

125

6.6 Regular Paths and Curves

Figure 6.6 Tangents to a smooth path.

D E FI N I T I O N 6.21.

A regular path is a smooth path γ : [a, c] → C such that γ ± (t)

for all t ∈ [a, b]. That is, every point on the path is a regular point. A regular curve is the image of a regular path.

²= 0

If γ is regular, then by Proposition 4.18 a point on the tangent at γ (t) is of the form γ (t) + hγ ± (t) for any h ∈ R, Figure 6.6.

The standard paths L(t) (line) and C(t) (circle) in Section 2.4 are regular. In Figure 6.6 the tangent line at γ (t) is a good approximation to the curve given by the image of γ , near that point. To formalise this idea, we compare the path γ (t) for t near some point t0 ∈ [a, b] with the corresponding tangent line. We can think of the tangent line as a path τ in its own right, deﬁned by

τ (t 0 + h) = γ (t0 ) + hγ ± (t0 )

and compare it with

γ (t0 + h)

(h ∈ R)

(6.12)

(t near t0 )

We now show that when h is small, these two paths, treating h as a parameter, are very close together for any given choice of h: P RO P O S I T I O N 6.22.

Let γ : [a, b]

→ C be a regular path. As h → 0, the expression 1 [γ (t0 + h) − τ (t0 + h)] h

tends to 0. Before giving the simple proof, we interpret this result: it says that for sufﬁciently small h, the difference between the path γ and the tangent path τ tends to zero faster than h does. Proof. Since γ is differentiable at t0 , lim

h→0

γ (t 0 + h) − γ (t 0) = γ ± (t ) h

0

126

Integration

Rewrite this as lim

h→0

1 [γ (t0 + h) − (γ (t0 ) + hγ ± (t0 ))] = 0 h

By (6.12) this is the same as lim

h→ 0

1 [γ (t 0 + h) − τ (t0 + h)] = 0 h

The proof does not use γ ± (t0) ² = 0; that is, the limit is still zero even when there is a singularity at t0. However, in that case τ (t0 + h) = γ (t0 ) for all h ∈ R. That is, τ does not deﬁne a tangent line. So the result tells us nothing about the shape of the image of γ near t0 .

6.6.1

Parametrisation by Arc Length Proposition 6.22 is a formal statement of the intuitive idea that a regular curve looks smooth near any point. It has a continuously turning tangent and a well-deﬁned ﬁnite length. These properties are inherited by subpaths, leading to: 6.23. Let γ : [a, b] → C be a regular path, with image curve C. Let t0 , t1 ∈ [a, b] with t0 ≤ t1, and let γ (t0 ) = c, γ (t1) = d. Then the arc length LC(c, d) from c to d in C is the length of γ| [t 0,t1] ; that is,

D E FI N I T I O N

L C(c, d)

=

²

t1

t0

|γ ± (t)| dt

We now prove that a regular curve can be smoothly reparametrised so that the parameter t is arc length, or a constant multiple of arc length if that is more convenient. Let the length of γ be L. Deﬁne λ : [a, b] → [0, L] by

λ(s) = LC (a, s) = Then

²

a

s

|γ ± (t)|dt

λ± (t) = |γ ± (t) | ² = 0

so λ is a strictly increasing function on [a, b] with a continuous derivative λ± on [a, b], where λ(a) = 0, λ(b) = L. It is regular since λ± (t) ² = 0. It therefore has a strictly increasing inverse function ρ = λ−1 . Now ρ : [0, L] → [a, b] and has continuous derivative

ρ± (t) = 1/λ± (t) ²= 0

for a < t < b, so ρ is also regular. The path

τ = γ ◦ ρ : [0, L] → C

is regular, and

τ (λ(t)) = γ ◦ ρ ◦ λ(t) = γ (t)

6.7 Regular and Singular Points

so τ is a reparametrisation of γ , with parameter λ(t) image curve C of γ by arc length. Moreover,

127

= LC (a, t). So τ reparametrises the

|τ ± (t)| = |γ ± (t)||ρ± (t)| = |γ ± (t)|/|λ± (t)| = |γ ± (t)|/|γ ± (t)| = 1 So as t varies, τ (t) moves along the path with unit speed.

Multiplying the parameter by a non-zero constant scales the speed 1 to any non-zero constant speed, corresponding to a parameter that is a constant multiple of arc length. Any curve parametrised by arc length or such a multiple is regular.

6.7

Regular and Singular Points Before tackling the intricacies of contour integration, we explain why the cases γ ± (t) ² = 0 and γ ± (t) = 0 differ signiﬁcantly. Near a regular point, the image of γ is a smooth curve in the geometric sense that it has a well-deﬁned tangent direction (and this varies continuously). The curve may cross itself, but each separate segment near the crossing looks smooth. Near a singular point, this may not be true. Sometimes there is a sensible, indeed visible, tangent direction, but sometimes there is not. The geometry of γ (t) when t is near a singular point t0 is highly sensitive to the precise behaviour of γ (t) when t is near t0 . We give a series of simple examples to illustrate some of the possibilities.

Figure 6.7 Examples of possible behaviour of a smooth path near a singular point. Left: Cusp.

Right: Right-angled corner.

γ −

γ = +

Example 6.24. Suppose that : [ 1, 1] C where (t) t2 it 3. The derivative is 2 (t) 2t 3it , which is non-zero except at t 0. The image is a semicubical parabola with a cusp at the origin, Figure 6.7 (left). Everywhere else, the image is a smooth curve. The image has a natural tangent line at the cusp point, along the real axis, because this is the limiting direction of tangents near the origin. However, the velocity vector reverses direction as it passes through the origin.

γ = +

=

128

Integration

To make sense of what is happening at a singular point we compare it with the real case. In real analysis the graph of a function f : R → R consists of the points (x, f (x)) ∈ R2 . If f is smooth, the graph has a well-deﬁned tangent – even at a critical point where f ±(x) = 0. However, if we imagine the point y = f (x) moving as x increases steadily, y becomes stationary at any critical point. (Indeed, another term is stationary point.) The tangent at a critical point is horizontal. This is the same as the direction in which a point (x, f (x)) on the graph is projected to obtain the image f (x). So the tangent projects to a single point. The same kind of behaviour is happening at the cusp, but now γ is a hybrid function with a graph γ : [ −1, 1] → C in coordinate form (t, t2 + it3 ) ∈ R × C. This is drawn in the upper left part of Figure 6.8, represented in three-dimensional real space as (t, x, y) = (t, t 2, t 3), where t, x, y all lie between −1 and +1. Projection onto the vertical (x, y)-plane

y = t3

3D (t, x, y)-space

3

project onto

y=t

vertical (x, y)-plane x = t2

0.5

x = t2

–1 0.5

–0.5

tangent –0.5

–0.5

(x, y)-plane

0.5 t

–1

–1 project down onto (t, x)- plane

pr oje ct on sid to ew (t , ay y )s pla ne

x = t2

y = t3

t

t

(t, x)-plane

(t, y)-plane

Figure 6.8 Example 6.24 represented in terms of the graph of the graph of the function (t) t2 it3 in pictured in 3 .

γ = +

R×C

R

129

6.7 Regular and Singular Points

reveals the semicubical parabola (t 2, t 3 ); projection down onto the (t, x)-plane gives the parabola (t, x2 ); and sideways projection onto the (t, y)-plane gives the cubic curve (t, x3 ). The tangent at the origin is in the direction (1, 2t, 3t 2) = (1, 0, 0), which points along the t-axis perpendicular to the complex plane and its projection onto the complex plane has length zero. Example 6.25. More surprisingly, the image can have a sharp corner, contrary to the intuition that it ought to at least look smooth. Deﬁne : R R by

φ

φ (t) =

µ

0 t2

(t (t

≤ 0) > 0)

This function is continuously differentiable because both 0 and t 2 have the same derivative, mainly 0, when t = 0. Now deﬁne γ : [−1, 1] → C by

γ (t) = φ(t) + iφ( −t)

The image is then as shown in Figure 6.7 (right). The deﬁnition of φ can be modiﬁed to make γ inﬁnitely differentiable, while retaining the right-angled corner. Just change t2 to e−1/t as in (4.8). Next we compare two different cases where the image of γ is a spiral. Example 6.26. Suppose that

γ (t) =

µ

0 tei/t

(t = 0) (t > 0)

Then γ : [0, 1] → C has a spiral image, Figure 6.9 (left). In this case, continuously differentiable. Indeed as t → 0 from above, we have

γ

is not

γ ± (t) = ei/t + t. ti ei /t = (1 + i)ei/t √ This goes round and round the circle of radius |1 + i| = 2 at an ever-increasing rate, so it does not tend to a limit as t → 0. The one-sided derivative at t = 0 is not deﬁned, let alone continuous.

Contrast this spiral with the following closely related one: Example 6.27. Suppose that

γ (t) =

µ

0 t 2ei/t

(t (t

= 0) > 0)

Then γ : [0, 1] → C has a spiral image, Figure 6.9 (right). The spiral is much tighter than the previous example, and now γ is continuously differentiable. Indeed as t → 0

130

Integration

Figure 6.9 Left: A spiral path that is not smooth. Right: A smooth spiral path.

from above, we have

γ ± (t) = (2 + i)tei/t

which tends to 0 as t → 0. The one-sided derivative at t = 0 is deﬁned, and the derivative is continuous there. As it happens, it is zero. We now consider the lengths of the spiral paths in Examples 6.26 and 6.27, which can be estimated by bounding the lengths of successive turns through 2π . In Example 6.26, turn n has length lying between two constant multiples of 1 /n. So the total length lies ∑ between the same constant multiples of n 1/n, the harmonic series, which is inﬁnite. In Example 6.27, turn n has length lying between two constant multiples of 1/n2. Since ∑ 2 n 1 /n is ﬁnite, this spiral has ﬁnite length. These results are consistent with the lack of smoothness of Example 6.26 (indeed, since its length is inﬁnite Proposition 6.11 implies that it cannot be smooth), and the smoothness of Example 6.27, which implies that its length must be ﬁnite.

6.8

Contour Integration For readers who took the Riemann integral route, Section 6.2 deﬁnes the notion of a smooth path and deduces a formula for the integral along such a path. Those who opted for the short cut should refer to Deﬁnition 6.15. For all readers, we now generalise the notion of integration to allow paths made up of a ﬁnite number of smooth pieces: D E FI N I T I O N

6.28. Using the notation of Section 2.4, a contour is a path of the form

γ = γ1 + · · · + γn where γ1, . . . , γn are smooth paths such that the ﬁnal point of γr coincides with the initial point of γr +1 for r = 1, . . . , n − 1.

6.8 Contour Integration

131

Figure 6.10 The contour deﬁned in Example 6.29.

Example 6.29. Let

γ1(t) = t (t ∈ [ε, R]) γ2(t) = R(cos t + i sin t) (t ∈ [0, π ]) γ3(t) = t (t ∈ [−R, ε]) γ4(t) = ε(− cos t + i sin t) (t ∈ [0, π ]) Then γ = γ1 + γ2 + γ3 + γ4 is the (closed) contour drawn in Figure 6.10. 6.8.1

Deﬁnition of Contour Integral Integration along a contour is an easy extension of integration along a smooth path: D E FI N I T I O N 6.30. If D is a domain, f : D → C is continuous, and γ = γ1 + · · · + γn is a contour (so all γr are smooth), then the contour integral of f along γ is

²

γ and

=

f

²

γ1

f

+ ··· +

²

γn

f

L(γ ) = L(γ1 ) + · · · + L( γn )

It is obvious that if a smooth path σ is subdivided as

σ = σ1 + σ 2 then

²

σ

f

=

²

σ1

f

+

²

σ2

f

so further subdivisions of the contours γ1 , . . . , γn in the above deﬁnitions will not affect the values of the integrals. The contour integrals are therefore well-deﬁned. The following standard properties hold, analogous to the real case:

132

Integration

P RO P O S I T I O N

6.31.

² ²

γ

f

=

( f1 + f2)

=

γ1+γ2 ²

γ ²

² ²γ1

γ²

cf

=c

f

=−

−γ

+

f

f1 +

²

f

γ ² γ

f

(6.13)

f2

(6.14)

(c ∈ C)

(6.15)

²γ2

γ

f

(6.16)

Proof. Equation (6.13) follows trivially from the deﬁnitions. The proofs of (6.14) and (6.15) depend on whether the reader has read Sections 6.1–6.3. If so, then Section 6.2 implies that S(P, f1 + f 2, γ ) = S(P, f 1, γ ) + S(P, f2 , γ ) S(P, cf , γ ) = cS(P, f , γ )

for any partition P, which imply (6.14) and (6.15). The reader who started from Section 6.8 can verify these formulas using known properties of real integrals, as follows. First, it is sufﬁcient to verify them for a single smooth path. To prove (6.14), observe that

²

γ

( f 1 + f2 ) =

= =

²

b

( f1 (γ (t)) + f 2(γ (t))γ ± (t)dt

b

f1 (γ (t))γ ± (t)dt +

a

²

a

²

γ1

f

+

² a

²

γ2

b

f2 (γ (t))γ ± (t)dt

f

using the additivity property for real integrals on the real and imaginary parts. The proof of (6.15) is slightly longer (a minor penalty for skipping Sections 6.1–6.3), because we must separate everything into real and imaginary parts. Let c = α + iβ , f (γ (t))γ ±(t) = U(t) + iV (t). Then

²

γ

cf

= = =

²

b

(α + iβ )(U(t) + iV (t))dt

b

([α U(T ) − βV (t)] + i[α V(t) + β U(t)])dt

a

²

a

²

a

b

²

[αU(T ) − βV (t)]dt + i

a

b

U(T )dt − β

²

a

b

²

a

b

[α V(t) + βU(t)])dt

V(t)dt + iα

²

a

b

V(t)dt + iβ

² a

b

U(t)dt

6.9 The Fundamental Theorem of Contour Integration

= (α + iβ) =c

²

² a

b

133

(U(t) + iV(t))dt

f

γ

using standard properties of real integrals. Finally, we prove (6.16). If γ : [a, b] −γ : [a, b] → D, deﬁned by

D is a contour, then the opposite path

−γ (t) = γ (a + b − t) is also a contour. Now

²

−γ

f

=

² a

=−

b

(t

f (γ (a + b − t))

²

b

a

∈ [a, b])

d γ (a + b − t)dt dt

f (γ (a + b − t))γ ± (a + b − t)dt

Substituting s = a + b − t this becomes

− =− =− Therefore

²

b

²

a

f (γ (s))γ ± (s)(−ds)

b

f (γ (s))γ ± (s)(ds)

²a

γ

f

²

−γ 6.9

f

=−

²

γ

f

The Fundamental Theorem of Contour Integration Integrals of complex functions are only occasionally computed by breaking them down into real and imaginary parts and calculating two real integrals as in the previous section. ± This technique is sometimes needed, but a far more efﬁcient method is available for γ f if we can ﬁnd an antiderivative of f . This concept is so important that we give it a formal deﬁnition: D E FI N I T I O N 6.32.

Let D be a domain. An antiderivative for a function f : D a function F : D → C such that F ± = f .

→ C is

An antiderivative, if it exists, is unique up to an added constant, for if F ± = G± = f in a domain, then (F − G)± = 0 so F − G is constant by Theorem 4.14. If we can ﬁnd an antiderivative, then the integral can be computed immediately using:

134

Integration

6.33 (Fundamental Theorem of Contour Integration). If f : D → C is continuous, F : D → C is an antiderivative of f , and γ is a contour in D from z0 to z1 , then

THEOREM

²

γ

f

= F(z1) − F(z0)

Proof. Let w(t) = u(t) + iv(t), W(t) = U(t) + iV(t) for t ∈ [a, b], where u, v, U, V are real. Then W± = w if and only if U ± = u, V ± = v, and then

²

b

²

b

²

b

= u(t)dt + i v(t)dt a a a = U(b) − U(a) + iV(b) − iV (a) = W(b) − W(a) Let w(t) = f (γ (t))γ ± (t). Since F ± = f , w ±(t) = F ± (γ (t))γ ± (t) = W ± (t) where W(t) = F(γ (t)). Therefore ²

γ

f

w(t)dt

=

²

a

b

w(t)dt

= W(b) − W(a)

= F(γ (b)) − F(γ (a)) = F(z1 ) − F(z0 )

One remarkable feature of this result is that, in these circumstances, the integral does not depend on the path γ , only on its end points. In contrast, we show in Section 6.10 that the integral may depend on the path when the conditions of Theorem 6.33 fail to hold.

=

γ

Example 6.34. If f (z) z2 and is any contour from z0 1 3 F(z) 3 z is an antiderivative of f , and

=

²

γ

z2 dz = 31 (z31 − z30 ) = 13 (1 + i)3

=

0 to z1

= 1 + i, then

= 32 + 32 i

This example gives a visibly easier calculation than that performed at the end of Section 6.2 for the particular contour γ (t) = t2 + it (t ∈ [0, 1]). However, any euphoria we feel over this phenomenon must be tempered with the realisation that, unlike the real case where Theorem 6.2 (iii) tells us that a continuous function always has an antiderivative, in the complex case there are continuous functions without antiderivatives. Indeed, as we prove in Chapter 10, any function that is differentiable once in a domain is differentiable as many times as we wish. Now, any antiderivative F must be differentiable once, so it is differentiable twice. Therefore its derivative f = F ± must be differentiable. So no non-differentiable function f has an antiderivative.

6.9 The Fundamental Theorem of Contour Integration

135

= ||

Example 6.35. We know that f (z) z 2 is differentiable only at the origin, so it is useless to search for an antiderivative when integrating this f – for example along the path (t) t 2 it (t [0, 1]). In this case, we return to the basic formula

γ = +

²

γ

²

|z|2dz =

1

0

²

(t4 + t 2)(2t + i)dt

²

1

1

= (2t 5 + 2t3 )dt + i (t4 + t2 )dt 0 0 = [ 31 t6 + 21 t4 ]10 + i[ 51 t5 + 13 t3 ]10 = 56 + 158 i Fortunately, many functions do have antiderivatives. For instance, by Corollary 4.5, any polynomial p(z) has antiderivative

= a0 + a1 z + · · · + an zn

p(z) = a0 z + 21 a1 z + · · · + n+1 1 anzn+1

More generally, a power series has an antiderivative everywhere inside its disc of convergence: THEOREM

6.36. If f (z) =

∑∞

n=0 a n(z

F(z) = also converges for |z − z0|

− z0)n converges for |z − z0 | < R, then

∞ a ³ n n=0

n+1

(z − z0 )n+1

< R, and F ± = f .

Proof. It is sufﬁcient to show that F(z) converges for |z − z0 | < R, for then we may differentiate term by term by Theorem 4.20 to obtain F ± = f . ∑ n We know that the power series ∞ n=0 a n(z − z0) converges absolutely for | z −z0 | < R, by Theorem 3.19. For each n, either an = 0 or

|an (z − z0 )n/ (n + 1)| = |z − z0| |an(z − z0)n | n+1 ∑∞

− z0) n+1 converges. ∑ n If f (z) = ∞ n=0 a n(z − z0 ) has disc of convergence D, then for any contour γ in D

tends to zero as n → ∞ , so by the comparison test from z1 to z2 ,

²

γ

f

=

∞ a (z − z )n+1 ³ n 2 0 n+1

n=0

∞ a (z − z )n+1 ³ n 1 0

In particular, for any contour γ in D from z0 to z,

²

γ

f

=

an n=0 n+1 (z

n=0

∞ a (z − z )n+1 ³ n 0 n=0

n+1

n+1

136

Integration

6.10

An Integral that Depends on the Path The Fundamental Theorem tells us that if a±differentiable function f has an antiderivative throughout its domain D, then the integral γ f is independent of the choice of path. This applies to zn for any integer n ² = −1, with antiderivative zn+1/ (n + 1) in the domain C\{0}. It applies to 1/z2 , 1/z3, . . . but fails for 1/ z, as we now show using formula (6.10).

/ /

Example 6.37. Consider the problem of integrating 1 z between 1 and 1. The path along the real axis passes through the origin, where 1 z is not deﬁned, so we avoid this by choosing a semicircular path. If the semicircle lies above the real axis (where im z 0) we can choose

γ1 (t) = e−it (t ∈ [π , 2π ]) noting that e−iπ = −1, e−i2π = 1 and the −t makes the path run clockwise. Now ²

²

2π 1 dz = γ1 z π

=

1 ± γ1(t) γ1 (t) dt

² 2π 1

π

² 2π

e−it

(−ie−it ) dt

= −i dt π = [−it]2ππ = −i(2π − π ) = −iπ All very well, but we could equally sensibly choose the semicircle that lies below the real axis (where im z ≤ 0). Now noting that ei π = −1, ei2π similar calculation yields

γ2 (t) = eit (t ∈ [π , 2π ]) = 1 and the +t makes the path run anticlockwise. A very ²

²

2π 1 dz = π γ2 z

=

1 ± γ2 (t) γ2(t) dt

² 2π 1

π

² 2π

eit

(ieit ) dt

= i dt π = [it]2ππ = i(2π − π ) = iπ which is slightly different.

6.11 The Gamma Function

137

At ﬁrst it might appear that we made a mistake – it’s easy to get a sign wrong – but the second integral is clearly the complex conjugate of the ﬁrst, because everything is reﬂected in the real axis. Since neither is real, they cannot be equal. This example does not contradict Theorem 6.33 because 1/z does not have an antiderivative in C\{0}. The obvious candidate is the logarithm, but by Section 5.7 exp is periodic over C hence not one-one. Therefore any inverse ‘function’ to exp must be multivalued, as explained in detail in Section 7.3. So the logarithm is not an antiderivative over the whole of the domain. In Chapter 8 we show that we get a different answer here because the point z = 0, where 1/z is undeﬁned, lies inside the region bounded by γ1 and γ2 . This behaviour is closely connected to the multivalued nature of the complex logarithm. It has surprisingly profound implications, explored in Chapters 8–12. At any rate, this example should counteract any feeling that complex integrals are always independent of the path, Theorem 6.33 notwithstanding.

6.11

The Gamma Function So far the special complex functions that we have encountered are all extensions to C of familiar real functions: polynomials, powers, trigonometric functions, the exponential and the logarithm. In this section we brieﬂy discuss a less familiar function, the gamma function ±(z). This also began as a real function and was later extended to the complex case. It has important applications, but we restrict attention to deﬁning the function and establishing a few basic properties. It will play a key role in Chapter 17 when we sketch connections between complex analysis and number theory, culminating in the Riemann Hypothesis. The deﬁnition of the gamma function goes back to Euler, who initially deﬁned it as the inﬁnite product

±(z) =

∞ »¶

1º z

1+

n=1

1 n

·z ¼

1+

z ½−1 n

¾

This inﬁnite product converges for all z ∈ C \{0, −1, −2 , . . .}. It deﬁnes a differentiable function on that domain, but we lack the techniques needed to prove this. Instead, we use another formula due to Euler, who showed that when re z > 0 this is equal to the integral

± (z) =

² ∞ 0

t z−1 e−t dt

(6.17)

We take this as our deﬁnition, in order to illustrate complex integration. Here we deﬁne

² ∞ 0

f

= alim →∞

²

[0,a]

f

so the interval [0, ∞ ] is interpreted as the non-negative real axis. It is straightforward to verify that (6.17) converges, because the factor e−t tends to zero rapidly. With this deﬁnition, the formula (6.17) deﬁnes ±(z) only for z in the positive half-plane

138

Integration

C+ = {z ∈ C : re z > 0}

We establish some basic properties of ±(z) using methods from complex integration. First, we prove that the usual method of integration by parts generalises from real analysis to the complex case: 6.38 (Integration by Parts). Let f , g : D → C be differentiable functions on the domain D. Then for any path γ : [a, b] → D we have:

THEOREM

²

γ

fg±

= [fg] ba −

²

γ

f ±g

(6.18)

Proof. By Proposition 4.4 (iii) we have (fg)± so

fg±

γ

Now integrate both sides over Integration, Theorem 6.33.

= f ± g + fg±

= (fg)± − f ±g

and use the Fundamental Theorem of Contour

Now we prove the basic functional equation for the gamma function:

∈ C+ then ±(z + 1) = z±(z) ±(1) = 1 By (6.17), noting that z + 1 ∈ C+ , we have

THEOREM

Proof.

6.39. If z

± (z + 1) =

² ∞ 0

(6.19)

t z e−t dt

Now use integration by parts to rewrite this as

± (z + 1) = [−t e−t ]∞0 + z

² ∞ 0

zt z−1 e−t dt

−x 0 = xlim →∞(−x e − (0e ) + z ² ∞ = 0 + z t z−1 e−t dt z

= z± (z)

To calculate ± (1), observe that

² ∞ 0

tz −1e−t dt

0

² ∞

±(1) = t 1−1 e−t dt 0 = [−e−t ]∞0 = 0 − (−1) =1 By induction, this formula leads immediately to a curious result:

6.11 The Gamma Function

C O RO L L A RY 6.40.

If n ∈ N then

139

±(n) = (n − 1)!

In effect, the gamma function is a sensible way to deﬁne factorials for complex numbers, in such a manner that the basic formula (n + 1)! = (n + 1)n! continues to hold.

6.11.1

Known Properties of the Gamma Function We record (but do not prove) several other properties of the gamma function. One is Euler’s reﬂection formula

±(1 − z)±(z) = sinππ z

which is a consequence of his inﬁnite product deﬁnition and a standard inﬁnite product for the sine. Setting z = 21 we obtain (±( 21 ))2 Therefore

= sinππ/2 = π

±( 21 ) = √π

The duplication formula states that

±(z)± (z + 21 ) = 21−2z √π±(2z)

The derivative of the gamma function is obtained from the integral formula (6.17) by a technique called ‘differentiating under the integral sign’, and leads to

± ± (z) =

² ∞ 0

t z−1 e−t log t dt

(see Chapter 7 for the complex logarithm). This also proves that function in C+ . Let γ be Euler’s constant, deﬁned by

γ = nlim →∞

1+

Then

1 1 1 + + · · · + 2 3 n

± is a differentiable

·

− log n ∼ 0.577 216

±± (1) = −γ

Weierstrass found another inﬁnite product that works for all z ∈ C \ { 0, −1 , −2, . . .}: 1 ± (z)

∞ ¿¼ º z ½ −z/ nÀ γ z 1+ = ze e n n=1

It is also possible to successively extend Euler’s formula to re z > −1, re z > −2, and so on, using (6.19) in the form

±(z − 1) = z±−(z)1

140

Integration

5

5

0

5 0

–5

0 –5

3

0.5

2

0

–0.5

–0.5

0

–5

0.0

0.0

1 5

–5 0.5

5

0 –5

–5

0

5

Figure 6.11 Graphs of (from left to right) the absolute value, real part, and imaginary part of the gamma function. Here x runs horizontally, y runs into the page, and the value of the function concerned runs vertically.

provided z − 1 ² = 0. This deﬁnes ± on the whole of the complex plane, except for the points {0, −1, −2 , . . .}, and shows that the extended function is differentiable on its domain. It also explains why negative integers and zero must be excluded from the domain, because ± would become inﬁnite there; that is, have a pole. This procedure is an example of analytic continuation, which will be introduced in Chapter 14. Graphs of |±(z)| , re ± (z), and im ±(z) are shown in Figure 6.11, for this extended domain. The gamma function, and its poles, turned out to be an important ingredient in a radical new development in complex analysis: an unexpected link with number theory. In the hands of Riemann and his contemporaries, this led to major advances in number theory, and a conjecture that remains one of the biggest unsolved problems in mathematics, the Riemann Hypothesis (see Chapter 17).

6.12

The Estimation Lemma We now return to more general questions about complex integration, and prove a very useful technical result. Some results in mathematics are servants of the theory, of little intrinsic interest in themselves, perhaps even a little dull, yet quietly ﬁguring in important decisions all the time. ‘Always a lemma, never a theorem’. One such result is the lemma we are about to prove. It is a simple idea, giving an upper bound for the size of ± an integral | γ f | in terms of an upper bound on | f | and the length of γ , but it arises time and again in the theory, applying subtle pressure at critical points in the proofs of important theorems. Perhaps not a theorem of great stature, it is certainly worthy of a name. It is: 6.41 (Estimation Lemma). If f : D length L, and | f (z)| ≤ M for all z on γ , then

LEMMA

´² ´ ´ ´ γ

→ C is continuous, γ

is a contour of

´ ´ f ´´ ≤ ML

Proof. It is poetic justice that we have to give two different proofs, depending on whether the reader has read Sections 6.1–6.3. Version A is the more natural, but it

141

6.12 The Estimation Lemma

requires knowledge of Sections 6.2–6.3. For readers who took the short cut, Version B is provided. Version A. It is sufﬁcient to prove the result for a smooth path γ : [a, b] → D. Let P be the partition a = t0 < t1 < · · · < t n = b, with tr −1 ≤ sr ≤ t r . Then

´ ´ n ´ ´³ ´ |S(P, f , γ )| = ´´ f (γ (sr ))(γ (tr ) − γ (tr−1))´´´

r =1 n ³ r =1

| f (γ (sr ))| |γ (tr ) − γ (tr−1)|

≤ ML(γP )

where γP is the polygonal approximation to L(γP ) ≤ L(γ ), so

γ

with vertices

γ (t0 ) , . . . , γ (tn).

|S(P, f , γ )| ≤ ML(γ )

But

(6.20)

for all partitions P. Given any ε > 0 we can ﬁnd a partition P ε such that

´ ´ f ´´ < ε

´ ² ´ ´S(P ε , f , γ ) − ´ γ

so

´² ´ ´ ´ γ

and (6.20) gives

for all ε

´ ´ f ´´ < |S(P ε , f , γ )| + ε

´² ´ ´ ´ γ

> 0. Hence

´ ´ f ´´ < ML(γ ) + ε

´² ´ ´ ´ γ

´ ´ f ´´ ≤ ML(γ )

as required. Version B. First we establish

´² ´ ² ´ b ´ b ´ ´ |u(t) + iv(t) |dt ´ (u(t) + iv(t))dt ´ ≤ ´ a ´ a ±b ±b

for continuous real functions u, v. Let

²

a

b

a

u(t)dt = X,

(u(t) + iv(t))dt

and X 2 + Y 2 = (X − iY )(X

=

²

a

b

+ iY ) =

a

v(t)dt

= Y. Then

= X + iY

²

b

a

(Xu(t) + Yv(t))dt + i

(X − iY)(u(t) + iv(t))dt

²

a

b

(Xv(t) − Yu(t))dt

(6.21)

142

Integration

But X 2 + Y 2 ∈ R, so

²

b

a

(Xv(t) − Yu(t))dt = 0

and X2 + Y 2

=

²

b

a

(Xu(t) + Yv(t))dt

where the integrand Xu(t) + Yv(t) is the real part of (X

− iY )(u(t) + iv(t)). So Xu(t) + Yv(t) ≤ |(X − iY)(u(t) + iv(t))| Á = X2 + Y 2|u(t) + iv(t)|

From real analysis,

²

a

b

[Xu(t) + Yv(t)]dt

which gives X so that

But

2

Á

+Y ≤ 2

Á

X2

² bÁ

X2

+

+ ≤ Y2

X 2 + Y 2 | u(t) + iv(t)| dt

a

Y2

² a

b

² a

b

|u(t) + iv(t)|dt

|u(t) + iv(t)|dt

(6.22)

´² ´ ´ b ´ ´ ´ X 2 + Y 2 = |X + iY | = ´ (u(t) + iv(t))dt ´ ´ a ´

Á

and with (6.22) this implies (6.21). Now

´ ´² ´ ´´² b ´ ´ ´ ´ ´ f ´ = ´ f (γ (t))γ ± (t)dt´´ ´ ´ γ ´ ´ a ² b

a

²

| f (γ (t))||γ ± (t) |dt

b

= |Mγ ± (t)|dt a = ML

by (6.21)

by real analysis

The Estimation Lemma will be used in a variety of ways. We now state two applications. P RO P O S I T I O N

6.42. Let γ be a ﬁxed contour and suppose that f varies.

±

(i) If the maximum value of | f | on γ tends to zero, then γ f → 0. ± ± More generally, if f → f0 , then γ f → γ f0 . (ii) If the length of γ tends to zero, the image of γ tends to a point z 0, and | f | remains ± bounded, then γ f → 0. In particular this is the case if f is continuous at z 0 .

6.13 Consequences of the Fundamental Theorem

143

Proof. To prove (i), observe that as the maximum value of | f | on γ tends to zero, ± ± γ f | ≤ ML, which also tends to zero, so γ f → 0. The second statements follows by considering f − f0 . To prove (ii), ﬁrst observe that if f is continuous at z0, then | f | is bounded in a neigh± bourhood of z0. So we may assume | f | is bounded on γ . Again, | γ f | ≤ ML, which tends to zero.

|

6.13

Consequences of the Fundamental Theorem If f is continuous and has an antiderivative F in a domain D, then we saw in Section 6.9 that, for any contour γ in D from z0 to z1,

²

γ

f

= F(z1) − F(z0)

This result has interesting consequences. To explore them we need: D E FI N I T I O N 6.43.

A contour γ : [a, b]

±a

→ C is a closed contour if γ (a) = γ (b).

In real analysis, an integral a f is always± zero. In the complex case, if f has an antiderivative in its domain D, then an integral γ f round a closed contour in D is always zero. (In the absence of an antiderivative, this need not be the case, as Example 6.37 shows.) Moreover, if f has an antiderivative in D and z0 , z1 in D are different, it does not matter which contour γ from z0 to z1 we use to integrate f : the result is always the same. These properties characterise the existence of an antiderivative: 6.44. Let f be a continuous complex function deﬁned on a domain D. Then the following conditions are equivalent: THEOREM

(i) (ii) (iii)

f± has an antiderivative in D, f = 0 for every closed contour γ in D, ±γ γ f depends only on the end points of γ for any contour γ in D.

Proof. We have already established that (i) implies (ii) and (iii). To show that (ii) implies (iii), suppose that γ1, γ2 are contours in D from± z0 to z1, as in Figure 6.12. Then γ1 − γ2 is a closed contour, so by (ii) γ −γ f = 0. But

²

so

f

γ1−γ2

=

²

γ1

f

1

²

f

γ1

=

=

γ2

f

²

γ2

which is (iii). Finally, to show that (iii) implies (i), we ﬁx z0 contour γ in D from z0 to z1 and deﬁne F(z1)

2

²

f

∈ D and for any z1 ∈ D we choose a

²

γ1

f

144

Integration

Figure 6.12 Two paths with the same end points determine a closed contour.

Because D is open, for some ε1 (t ∈ [0, 1]) lies in D, and

> 0, if |h| < ε1 then the line segment λ(t) = z1 + ht

F(z1 + h) =

Thus

+

f

=

γ from z1 to z2, ²

γ

Therefore

γ1

F(z1 + h) − F(z1 ) h

For any constant c and contour

so

²

c dz

²

λ

f

²

1 f h λ

= c(z2 − z1)

²

f (z1 ) dz = f (z1) λ h

F(z1 + h) − F(z1) h

²

f (z) − f (z1 ) dz h λ Now we use the Estimation Lemma. Continuity of f implies that, given exists δ > 0 (which we may take to be less than ε1 ) such that

− f (z1 ) =

ε > 0, there

|z − z1 | < δ implies | f (z) − f (z1)| < ε so that, when 0 < |h| < δ, for any z on the line segment λ the integrand satisﬁes |( f (z) − f (z1))/h| < ε/|h| The length of λ is |h|, so whenever |h| < δ, ´ ´² ´ f (z) − f (z1 ) ´ ε |h| = ε ´ dz´´ ≤ ´ h |h| λ which implies

´ ´ ´ ´ F(z1 + h) − F(z1 ) ´ − f (z1)´´ ≤ ε (|h| < δ) ´ h

6.13 Consequences of the Fundamental Theorem

But ε is arbitrary, so

F(z1 + h) − F(z1) h h→ 0 lim

and F ± (z1) = f (z1 ) for z1

145

= f (z1 )

∈ D, as required. = ||

±

Example 6.45. The function f (z) z is not differentiable, so γ f may (and as we show, does) depend on the choice of contour between points. For instance, consider , from 0 to i given by

γσ

and σ

γ (t) = it (t ∈ [0, 1])

= σ 1 + σ2 where

as in Figure 6.13. Then

σ1(t) = t (t ∈ [0, 1]) σ2(t) = eit (t ∈ [0, π/2]) ²

γ but

²

σ

|z|dz =

² 0

1

|t | · 1 dt +

Figure 6.13 Two paths

γ

|z|dz = ² π/2 0

² 0

1

|it | · idt = Â

Â

it 2/2

Ã

Ã1 0

|eit | · ieit dt = t2 /2 10 +

= 21 i Â it Ãπ/2 e

0

= 12 + i − 1 = i − 21

γ and σ1 + σ2 that yield different values of the integral of |z|.

While complex functions that are merely continuous may be integrated using the basic formula, they offer little of interest. From now on, differentiable functions, and methods for integrating them, occupy centre stage. The resulting theory of integration has no natural real counterpart, because the real line lacks the ﬂexible choice of path between points that occurs in the complex plane. With minor exceptions, this chapter completes the natural analogies between the real and complex theories of differentiation and integration. From now on, new possibilities will unfold.

146

Integration

6.14

Exercises 1. Draw the contours γ

2. Draw the contours γ

±

= [0, i], σ = [0, 1] + [1, i]. Evaluate ²

γ

²

re z dz

σ

re z dz

= [−i, i] and σ where σ (t) = eit (t ∈ [−π/2, π/2]). Evaluate ² ² |z|dz |z|dz γ

3. Compute γ z dz for γ (t)

σ

= (1 + i)t (t ∈ [0, 1]) using the formula

4

²

γ

f (z)dz =

²

1

0

f (γ (t))γ ± (t)dt

and multiplying out the right-hand side and integrating the real and imaginary parts. ± If γ = [0, 1] + [1, 1 + i], what is γ z4dz?

4. Let n be a positive integer. If γ (t) and −γ geometrically. Compute

²

1

γ z − z0

= z0 + reit (t ∈ [0, 2nπ ]) describe the contours γ ²

dz and

1

−γ z − z0

±

dz

5. For γ (t) = eit (t ∈ [0, π ]) evaluate γ f for each of the following functions f : (i) 1/z2 (ii) 1/z (iii) cos z (iv) sinh z (v) tan z (vi) (exp(z))3 6. Verify that the length function L satisﬁes L(−γ ) = L(γ )

where γ , σ are contours.

L(γ

+ σ ) = L(γ ) + L(σ )

7. Let γ (t) = z0 + reit (t ∈ [0, θ ]) be the arc centre z0 , radius r, subtending angle θ . Using Proposition 6.11, verify that

8.

9.

θ = L(γ )/r which is the usual deﬁnition of ‘angle θ in radian measure’. Show that the length of the parabolic arc γ (t) = at2 + 2ait (t ∈ [0, 1]), where 0 < a ∈ R, is √ )) ( (√ a 2 + log 1 + 2 If γ is a closed contour in C, the signed area enclosed by γ is deﬁned to be ² 1 S= z¯ dz 2i γ

6.14 Exercises

147

Figure 6.14 What does the signed area represent for this contour?

±b

By writing the integral explicitly in the form a (u − iv)(u± + iv± ), or otherwise, show that S is real. Show that its value for circular and triangular contours is ³ the usual area, and that both signs can occur. What is the geometric signiﬁcance of the sign? What does the signed area represent for a contour such as Figure 6.14? 10. Show that the signed area in Exercise may also be given by S=−

²

γ

im z dz

±

S=

²

1 re z dz i γ

Write down the value of the integral γ im z, where [1, 1 + i] + [1 + i, i] + [i, 0].

11. Let γ (t) = eit (t

± (i) γ z sin z dz ± (ii) γ z cos z dz ± (iii) γ zeiz dz ± 2

γ

is the square

γ = [0, 1] +

∈ [0, π/2]). Use integration by parts, Theorem 6.38, to compute

(iv) γ z sin z dz 12. Let cr (t) = z0 + reit (t ∈ [0, 2π ]) be the circle centre z0 , radius r > 0. If f is continuous in the domain D and z0 ∈ D, use the Estimation Lemma to prove (i) lim

²

r →0 Cr ²

f (z)dz = 0

f (z) dz = 2π if (z0) r →0 Cr z − z0 13. For each of the following functions f : D → C, either state an antiderivative F : D → C or explain why an antiderivative does not exist. (ii) lim

(i) (ii) (iii) (iv) (v) (vi)

f (z) = z2 , D = C f (z) = 1/z2, D = C \ { 0} f (z) = 1/z, D = C \ {0} f (z) = z sin z, D = C f (z) = | z|2, D = C f (z) = z¯, D = C

148

Integration

x! 20 10 –3½ –4½

–4

–3

–2½

–2

–1½

–1

–½

½

1

2

3

4

x

–10 –20

!

Figure 6.15 Graph of x for real x.

14. Use Euler’s reﬂection formula

± (z)±(1 − z) = sin(ππ z)

to ﬁnd ±( 21 ). Use the identity ±(z + 1) = z±(z) to calculate ± (n + 21 ) for all n ∈ N. Gauss’s pi function has the value ²(z) = ±(z + 1) and is known to be analytic in D = C \ {n ∈ Z : n < 0}. By translating the identity ± (z + 1) = z±(z) to ²(z) = z²(z − 1) show that ²(0) = 1 and ²(n) = n! for n ∈ N. Figure 6.15 shows the graph of x! for real values of x. Calculate the numerical values of (n + 12 )! for n = 0, 1, 2, 3, −1, −2, −3, −4, and mark them on the graph.

7

Angles, Logarithms, and the Winding Number

So far, we have deﬁned complex analogues of most of the basic special functions of real analysis – the exponential and trigonometric functions – and proved that many of their familiar properties extend to the complex case. Some new features also arise, such as the link between exponential and trigonometric functions and the periodicity of exp. There is one glaring exception to our catalogue of complex analogues: the logarithm. In real analysis, we deﬁne this as the inverse function to exp : R → R+ , so log : R+ → R. However, because exp is periodic in C, it is not one-one (injective). Therefore it does not have an inverse function in the strict set-theoretic sense. This feature caused many of the historical disagreements about logarithms of negative and complex numbers, exacerbated by the general view that log z is automatically meaningful when z ∈ C, and all of its properties ought to generalise to C. Resolving these issues involves being far more careful when deﬁning complex logarithms. But the effort is worthwhile, because logarithms hold the key to many of the deeper and more powerful features of complex analysis. The problem of ﬁnding inverses to functions that are not bijections arises in real analysis as well. Inverse trigonometric functions such as sin −1 are notorious for causing severe headaches, but the square root is the most obvious case. The square f (x) = x2 is not injective as a function f : R → R. It is also not surjective: negative real numbers are not squares of real numbers. The historical resolution was to deﬁne two square roots for √ √ every positive real number, ± x. Each is a one-sided inverse: (± x)2 = x when x > 0. But neither is an inverse on the domain of f , which is all of R: their images are the positive reals and the negative reals, respectively, so they are not surjections onto R. To obtain genuine inverse functions, we restrict the domain of f to the non-negative reals, √ √ when x is an inverse, or to the non-positive reals, when − x is an inverse. These two domains meet only at the origin. It would be natural to try the same trick in the complex case. But now there is no very natural way to restrict the domain and codomain to make the logarithm a bijection. A variety of more or less artiﬁcial choices exist, such as the ‘cut plane’ Cπ where the negative real axis is deleted, and these have their uses. To obtain a natural description of the complex logarithm, though, we use a more geometric approach. In classical terms, the complex logarithm is ‘multivalued’ – its value is not uniquely deﬁned. The way in which its multiplicity of values ﬁt together is closely analogous to the way that the measurement of an angle in radians gives not a single real number, but an inﬁnite list differing only by multiples of 2π . Indeed, these two instances of non-uniqueness are closely

150

Angles, Logarithms, and the Winding Number

related. In particular, the well-known complications when deﬁning inverse trigonometric functions in real analysis also arise because these functions are 2π -periodic, hence not injective. The same issue has to be resolved to make sense of the complex logarithm. We discuss these ideas below, and apply them to a topological invariant known as the winding number of a path relative to a point z0 ∈ C. In essence, this measures the total angle relative to z0 traversed by a point moving along the path continuously from beginning to end. When divided by 2π , this angle tells us how many times the path winds round the point z0 . The winding number proves to be extremely useful in the deeper parts of the subsequent theory.

7.1

Radian Measure of Angles We ﬁrst relate the power series deﬁnitions of sine (5.7) and cosine (5.8) to the usual geometric ones, with the angle being measured in radians. This completes the proof that the power series represent the traditional trigonometric functions in the real case. Let 0 ² = z ∈ C and write z in polar coordinates as z = rei θ , (r > 0). Since eiθ = cos θ + i sin θ , Table 5.1 in Section 5.5 implies that for ﬁxed r, as θ increases from 0 to π/2, the real part of z decreases monotonically from r to 0 while the imaginary part increases monotonically from 0 to r. Since |z| 2 = r 2(cos 2 θ + sin2 θ ) = r2 , it follows that z traces out the ﬁrst quadrant of a circle of radius r (in the anticlockwise direction) as θ increases from 0 to π/2, Figure 7.1 (left). Similarly, as θ increases from π/2 to π , the point z traces out the second quadrant of a circle of radius r; from π to 3π/2, it traces out the third quadrant; from π to 3π/2, it traces out the fourth; and thereafter it continues to go round the circle in the anticlockwise direction, Figure 7.1 (middle). We now compute the arc length from 1 to z along the circular path. For θ ≥ 0, let γ (t) = reit , (t ∈ [0, θ ]). Then γ is a contour from 1 to z along the relevant arc. The length of γ is

Figure 7.1 Left: reiθ traces out the ﬁrst quadrant of a circle of radius r as increases from 0 to 2. Middle: For , re iθ repeatedly traces out the circle of radius r in the anticlockwise

π/

θ ∈R

θ

direction. Right: Traditional geometric deﬁnitions of sin θ and cos θ agree with the power series deﬁnitions.

7.2 The Argument of a Complex Number

L(γ ) =

± θ 0

|γ ³ (t)|dt =

± θ 0

|ireit |dt =

± θ 0

rdt

151

= rθ

Thus θ = L(γ )/r, which is the standard deﬁnition of ‘angle measured in radians’. Figure 7.1 (right) then shows that the geometric deﬁnitions of sin θ and cos θ , in terms of a right triangle with an angle θ radians, agrees with our analytic deﬁnition via power series. The 2 π -periodicity of sin and cos proved in Proposition 5.5 shows that this agreement extends to angles greater than 2π , and to negative angles, which of course are measured in the opposite sense round the circle: clockwise.

7.2

The Argument of a Complex Number We now look in more detail at the expression of a complex number z in the form reiθ . Here r = |z|, so it is unique. However, when r = 0 any value of θ leads to z = 0. And when r > 0 there are inﬁnitely many possible values of θ , differing only by integer multiples of 2 π . Recall that the unique value in the range −π < θ ≤ π is called the principal value of the argument of z, and is denoted by arg z This deﬁnes a function

arg : C \ {0} :→ R

The function arg is not continuous on the negative real axis. This is a result of the need to choose θ uniquely: just above the axis θ is close to π , just below it is close to −π . We can sidestep this problem (inelegantly) by deﬁning the cut plane

Cπ = C \ {x + iy : y = 0, x ≤ 0} as in Figure 7.2. We claim that arg is continuous in the cut plane. This is plausible geometrically, but we must provide a rigorous proof. There is one technical difﬁculty: the bad behaviour of

Figure 7.2 The cut plane

Cπ .

152

Angles, Logarithms, and the Winding Number

inverse trigonometric functions (also caused by periodicity). We circumvent it by using several overlapping domains, on each of which the behaviour is sufﬁciently good. The proof that follows is an inelegant ‘bare hands’ reduction to properties of real functions: for a more elegant approach see Section 8.4. P RO P O S I T I O N

7.1. The function arg is continuous in the cut plane Cπ .

Proof. Let

= {x + iy ∈ C : y > 0} D2 = {x + iy ∈ C : x > 0} D3 = {x + iy ∈ C : y < 0}

D1

Then

C = D1 ∪ D2 ∪ D3

(draw a picture!) and in each of these domains we have an easy way to ﬁnd the value of arg z. To do so, observe that if and we wish to solve for r, θ , then r2

z = x + iy = eiθ

= x2 + y2

so r

=

²

x2 + y2

and

= ³ 2x 2 x +y y sin θ = ³ x2 + y2 From properties of sin and cos it follows that if z ∈ D1 there is a unique solution for θ with 0 < θ < π . In this range cos is monotonic strictly decreasing and continuous, so its restriction to (0, π ) has a continuous inverse function cos−1 : (−1, 1) → (0, π ) For z ∈ D1 we have re z arg z = cos−1 |z| cos θ

which, being a composition of continuous functions, is continuous. Similarly if x ∈ D2 then −π/2 < θ < π/2. In this range, sin increases monotonically from −1 to 1, so it has a continuous inverse sin−1 : ( −1, 1) → Now

which is continuous on D2.

´

arg z = sin−1

−π , π µ 2

im z | z|

2

7.3 The Complex Logarithm

Figure 7.3 The cut plane

153

Cα .

Finally, on D3 we can use

cos−1 : ( −1, 1) → (−π , 0)

which is a different choice of inverse cosine from that used in D1 because we are inverting the restriction of cos to a different interval. Again arg is continuous on D3. Since D1 , D2, D3 are open, arg is continuous in D1 ∪ D2 ∪ D3 = Cπ . It is sometimes convenient to proceed more generally. Let ray Rα Let

= {reiα : r ≥ 0} ⊆ C Cα = C \ Rα

as in Figure 7.3. Choose by the rule

α ∈ R, and let Rα be the

θ = arg α z z = reiθ , r

(z ∈ Cα )

> 0, α − 2π < θ < α

Then by a similar method it may be shown that arg is continuous in Cα .

7.3

The Complex Logarithm The complex logarithm poses similar problems, because the complex exponential function is not one-one. We wish to deﬁne log z, for 0 ² = z ∈ C, by w = log z if z = ew

(We exclude z = 0 because ew is never zero, by Proposition 5.4.)

154

Angles, Logarithms, and the Winding Number

Let z = reiθ , w = u + iv. (The mixture of polar and Cartesian coordinates is deliberate!) We assume r > 0, −π < θ ≤ π , so θ = arg z. Then z = ew becomes reiθ

Taking moduli,

= eu+iv

(7.1)

r = eu

(7.2)

Since r > 0 and r, u ∈ R this has the unique solution u = log r

where log is the real natural logarithm. Then (7.1) and (7.2) imply that eiθ so that

v=θ

Hence log z or

= eiv

+ 2nπ

(n ∈ Z)

= w = log r + i(θ + 2nπ )

log z = log | z| + i(arg z + 2nπ )

(7.3)

The complex logarithm is thus ‘multivalued’ and therefore not a function in the strict set-theoretic sense. To obtain a genuine (single-valued) function we deﬁne the principal value of the logarithm to be Log z = log | z| + i arg z (0 ² = z ∈ C) (Note the capital ‘L’ for the principal value.) This function is not continuous on the negative real axis. However, for z ∈ Cπ the real and imaginary parts of Log are clearly continuous, so Log is continuous in the cut plane. Now we can compute the derivative of the logarithm. We have

setting z

= ew , z0 = ew

Log z − Log z0 d Log z0 = lim z → z dz 0 z − z0 w − w0 = wlim →w0 ew − ew0

0

and using continuity of Log. Thus

1 d dz

and

Log z0

= wlim →w

0

ew − ew0 w − w0

d Log z0 dz

= dzd exp w0 = ew

= ew1 = z1 0

0

and in general 1 d Log z = dz z

(z ∈ Cπ )

0

7.4 The Winding Number

155

In the same way, in the cut plane Cα we can deﬁne logα z

= Log |z| + i argα z

and this is continuous, with derivative d 1 log α z = dz z

(z ∈ Cα)

Once the logarithm is deﬁned, powers za for complex a can be deﬁned, in a cut plane, by reducing them to exponentials. The principal value of za, when z ² = 0, is za

= exp(a Log z)

Exercises 9–14 of Section 7.9 explore this function. For a more global view of za see Chapter 14.

7.4

The Winding Number Suppose that γ : [a, b] → C \{ 0} is a closed path. The choice of codomain here ensures that the path does not pass through the origin. If we imagine t, the parameter, to be time, increasing from a to b, and choose the argument θ (t) of γ (t) to vary continuously with t, then as γ (t) moves along the path, then intuitively the total change in argument is the number of times that γ winds round the origin (anticlockwise), multiplied by 2π . (If γ is not closed this can also be deﬁned, but it need not be an integer.) We wish, of course, to make this idea precise: A continuous choice of argument along a path γ : [a, b] → C \ {0} (which for full generality we no longer require to be a closed path) is a continuous map θ : [a, b] → R such that γ (t) eiθ (t) = (7.4) |γ (t) | D E FI N I T I O N 7.2.

Condition (7.4) merely says that θ (t) is one of the possible values of the argument of γ (t). Before proving that a continuous choice of argument exists, in Theorem 7.4 below, we consider a simple example.

γ =

∈ π

Example 7.3. Let (t) reit (t [0, ]), Figure 7.4. A continuous choice of argument is (t) t, because obviously reit reit eit reiθ (t) . But this is not the only possible choice: (t) t 2 works equally well, or indeed (t) t 2n for any ﬁxed n Z. What we cannot do is ‘change horses in midstream’, say by choosing

θ = + π

θ =

θ (t) =

t t + 2π

(t (t

θ = + π

∈ [0, π/2]) ∈ (π/2, π ])

If we do that, condition (7.4) holds, but θ is not continuous.

/| | =

=

156

Angles, Logarithms, and the Winding Number

Figure 7.4 A semicircular path.

The difﬁculty (though it does not take much getting used to) in handling continuous choices of argument is that it does not in general sufﬁce to insist on some simple recipe for choosing a value of the argument, such as keeping angles between 0 and 2π . Such a choice becomes discontinuous if the path winds round once anticlockwise, taking the argument to 2π , and then continues in the anticlockwise direction. Then the argument jumps discontinuously back to near 0 when we want it to continue increasing from 2π . It turns out that we can choose the argument at the starting point to be any of the inﬁnitely many possible values; but this choice then determines all of the rest uniquely – and the arguments that arise need not stay within any predetermined range. It is intuitively obvious that this kind of choice can be made, and that it is unique once we decide where to start. Its proof is a little tricky: it uses the Paving Lemma. 7.4. (i) Let γ : [a, b] → C \ {0} be a path not passing through the origin. Then there exists a continuous choice of argument for γ . (ii) Any other continuous choice of argument differs from this by a constant integer multiple of 2π . THEOREM

Proof. By the Paving Lemma 2.33 we can subdivide γ into ﬁnitely many subpaths γr (r = 1, . . . , n) such that each γr lies inside a disc Dr ⊆ C \ {0}, Figure 7.5. If the centre of dr is at ρ r eiθ , then taking αr = θr + π , we ﬁnd that Dr ⊆ Cα , so argα is r

continuous in Dr . This gives a continuous choice of argument across γr for each r, but of course these choices need not ﬁt together continuously. However, any choice of argument can be obtained from any other by adding a suitable integer multiple of 2π . We can therefore adjust the argαr inductively so that they do ﬁt together continuously, as follows. As usual, let γr be deﬁned on the parametric interval [tr −1, tr ]. There is an integer n2 such that arg α1 (t 1) = arg α2 (t 1) + 2n2 π Then there is an integer n3 such that argα2 (t2 ) + 2n2 π

= argα (t2 ) + 2n3π 3

7.4 The Winding Number

157

Figure 7.5 Paving a path.

and so on inductively, choosing nr +1 so that argαr (tr ) + 2nr π

= argα + (tr) + 2nr+1 π r 1

Then we deﬁne

θ (t) = argα (tr ) + 2nr2π r

(t ∈ [t r−1 , tr ])

with n1 conventionally deﬁned to be 0. Then θ is continuous. This proves that a continuous choice of argument exists. Next, suppose that θ ∗ is another continuous choice of argument. Then we must have

θ ∗ (t) = θ (t) + 2n(t)π where n(t) is an integer, possibly depending on t. But n(t)

∗ (t) − θ (t) 2π

is continuous, yet takes integer values. It is therefore constant. Note that a continuous choice of argument is deﬁned on the parametric interval [a, b], not on the image of γ . This means that if the path returns to the same point, with γ (t1 ) = γ (t 2) but t1 ²= t2, the arguments θ (t 1) and θ (t2 ) may be different. Intuitively, it is clear that this will happen if the path winds round the origin a non-zero number of times between t1 and t2 . Example 7.5. Let (t) re4π it (t [0, 1]). A continuous choice of argument is given by (t) 4 t 2n , for any choice of 1 1 n Z. Although (t) (t 2 ) for t [0, 2 ], the choices of argument (t) and (t 21 ) differ by 2 , because the path has travelled once round the origin (anticlockwise) between t and t 12 .

γ =

π

γ =γ +

+

θ = π + π

θ

θ +

158

Angles, Logarithms, and the Winding Number

More signiﬁcantly, γ (0) = γ (1), but the difference in arguments is 4π , because in total the path has travelled twice round the origin (anticlockwise) between t = 0 and t = 1. We can take advantage of this idea: D E FI N I T I O N

is

7.6. The winding number of a path γ : [a, b] → C \ { 0} round the origin

θ (b) − θ (a) 2π for a continuous choice of argument θ along γ . w(γ , 0) =

By Theorem 7.4 (ii) the winding number is well deﬁned; that is, any continuous choice of argument gives the same value. For any closed path, the winding number is an integer, because θ (b) − θ (a) is an integer multiple of 2 π . The winding number takes the sense of the path into account: anticlockwise turns count as positive, clockwise turns count as negative.

γ =

π θ = −t. Then θ (6π ) − θ (0) = −3 w(γ , 0) = 2π

Example 7.7. Let (t) re−it (t [0, 6 ]). A continuous choice of argument is (t)

and γ winds three times round the origin in the clockwise direction. The winding number is additive: 7.8. Let γ1 and γ2 be two paths in C \ {0} such that the end point of γ1 is the start of γ2 . Then

THEOREM

w( γ1 + γ2 , 0) = w(γ1, 0) + w(γ2, 0)

Proof. We may assume that γ1 , γ2, and γ1 + γ2 have parametric intervals [a, b], [b, c], and [a, c] respectively. Let θ be a continuous choice of argument on γ1 + γ2 . Then

= θ (c) 2−π θ (a) θ (b) − θ (a) w(γ1, 0) = 2π θ (c) − θ (b) w(γ2, 0) = 2π

w(γ1 + γ2, 0)

This is an extremely useful result, because it lets us compute the winding number of a complicated path by breaking it up into nice pieces and adding their contributions. It extends easily to show that w(γ1 + · · · + γn , 0) = w(γ1, 0) + · · · + w(γn , 0) and that

w(−γ , 0) = −w(γ , 0)

7.6 The Winding Number Round an Arbitrary Point

7.5

159

The Winding Number as an Integral If the path is smooth, we can specify its winding number as an integral along the path. Additivity of the winding number extends this formula to contours. First, consider a closed contour. THEOREM

7.9. Let γ be a closed contour. Then w(γ , 0) =

±

1 1 dz 2π i γ z

(7.5)

Proof. Subdivide the path into subpaths γ1 , . . . , γn as in Theorem 7.4, whose notation we now use. Each γr lies in a cut plane Cαr . If γr is deﬁned on the interval [tr −1 , tr ] then

±

1 dz = log αr γ (tr ) − logαr γ (tr −1) γr z = log |γ (tr)| − log |γ (tr−1 )|

= i(argα |γ (tr)| − arg α |γ (t r−1 )|) As in Theorem 7.4 we make sure that arg α γ (tr ) = argα + γ (tr ), that is, the choices r

r

r

r 1

of argument agree where the subpaths join together. Then, summing the integrals for r = 1, . . . , n, the real parts cancel out because γ is a closed contour, and the imaginary parts add up to 2 π w( γ , 0), as required.

7.6

The Winding Number Round an Arbitrary Point There is nothing very special about the origin as far as winding numbers are concerned. If γ : [a, b] → C is a path and z0 ∈ C does not lie on γ , we can deﬁne the winding number of γ around z0. The easiest way to do this is to translate the origin, setting

±(t) = γ (t) − z0

and calculating

∈ [a, b])

w(γ , z0 ) = w(±, 0)

For a closed path γ , we get w(γ , z0) =

=

±

1 dz 2π i ± z ± b ³ ± (t) dt 1 2π i a ±(t) 1

= 2π1 i 1

= 2π i This leads to:

(t

±

±a

b

γ ³ (t) dt γ (t) − z0

1 dz z − z0 γ

160

Angles, Logarithms, and the Winding Number

7.10. If γ : [a, b] → C is a closed path and z0 then the winding number of γ around z 0 is

D E FI N I T I O N

w(γ , z0)

1

= 2π i

±

1

γ z − z0

∈ C does not lie on γ ,

dz

(7.6)

The additivity of w(γ , 0) in Theorem 7.8 extends easily to w(γ , z0 ) for ﬁxed but arbitrary z0 ∈ C, using the same trick of translating the origin.

7.7

Components of the Complement of a Path We consider how the winding number w(γ , z0) of a given closed path γ can vary as z0 varies. By Proposition 2.41, the complement S of the image of γ is open, and also each connected component of S is open. The winding number w(γ , z0 ) is deﬁned for all z0 ∈ S, and it is an integer-valued function on S. It is geometrically plausible that this function is constant on any connected component of S. We prove this analytically by showing that w(γ , z0 ) is a continuous function of z0 ∈ S. The desired result then follows, since any integer-valued continuous function is constant on any connected set. The proof that w(γ , z0 ) is continuous in z0 is obtained by a direct estimate. Fix z0 ∈ S. Since S is open there exists k > 0 such that | z1 − z0 | < k implies z1 ∈ S. Therefore if z lies on the image of γ then | z − z0| ≥ k. Thus if |z1 − z0| < k/2 we have | z − z1| > k/2. Now ·± ¸ ¹ ·· 1 1 ·· 1 |w(γ , z0 ) − w(γ , z1 )| = 2π · z − z − z − z dz·· 0 1 γ

·

·

· ·± = 21π ·· (z −zz1 −)(zz−0 z ) dz·· 0 1 γ | z1 − z0 | ≤ π k2

by the Estimation Lemma 6.41. Given any ε > 0, take δ = min(k/ 2, π k2 ε/2L(γ )). Then

|z1 − z0 | < δ implies |w(γ , z0 ) − w(γ , z1 )| < ε

Hence w(γ , z0 ) is continuous in z0 on S.

Example 7.11. The path in Figure 7.6 has the winding numbers shown, around points z0 in the components of S to which those numbers are assigned. The set S has a single unbounded component (Proposition 2.41) which we denote by O( ) (where the ‘O’ stands for ‘outside’). As Figure 7.6 shows, if z0 O( ) then w( , z0 ) 0. This is easily proved from the integral formula (7.6), as follows. Let z0 be ‘far from the image of ’, that is, assume z z0 K for large K. Then the Estimation Lemma 6.41 shows that

γ

∈ γ

| − |≥

γ

=

|w(γ , z0 )| ≤ L(γ )/2π K

γ

7.8 Computing the Winding Number by Eye

161

Figure 7.6 Winding numbers for components of the complement of a path.

which tends to zero for large K. But the left-hand side is a non-negative integer, so it must be equal to zero for large enough K. Since it is constant on O(γ ), it must be zero on the whole of O(γ ).

7.8

Computing the Winding Number by Eye The somewhat complicated deﬁnition of the winding number may give the impression that it is complicated to calculate. This is not so, at least for paths that are contours made up of smooth subpaths. For contours the calculation is simple: trace your ﬁnger along the contour and count the number of turns. The point of this section is that this process can easily be formalised so that, in this case, what is obvious is also true. Example 7.12. We start with a path Figure 7.7.

γ whose image is a rectangle with vertices ±2 ± i,

Figure 7.7 A rectangular path.

If you want a formula, it is not hard to give one; for example let

⎧ 2 − i + 2it ⎪ ⎪ ⎨ 2 + i − 4(t − 1)

γ (t) = ⎪ −2 + i − 2i(t − 2) ⎪ ⎩ −2 − i + 4(t − 3)

(t (t (t (t

∈ [0, 1]) ∈ [1, 2]) ∈ [2, 3]) ∈ [3, 4])

162

Angles, Logarithms, and the Winding Number

This path is composed of four standard ‘line segment’ paths, one for each edge, Section 2.4. As the parameter t varies, the corresponding point moves along each edge at constant speed. (With our deﬁnition of a line L(t), the speed along an edge depends on the length of that edge because t is taken in the interval [0, 1].) We want to calculate w(γ , 0). First, here is how not to do the calculation. Break γ up in the most natural way, into subpaths γr = γ| [r−1,r] for r = 1, 2, 3, 4. Then use the integral formula w(γ , 0) =

±

Now (to take one subpath)

±

1 dz γ1 z

4

±

1 1 1 º 1 dz = dz 2π i γ z 2π i γr z r=1

±

γ ³ (t) dt 0 γ (t) = [log(2 − i + 2i)]10 = Log (2 + i) − Log (2 − i) =

1

because γ1 lies in Cπ , so the principal value Log is continuous on γ1. Then Log (2 ± i) = log | 2 ± i| + i arg (2 ± i)

= log

so

5 ± i sin−1 (1/ 5)

(7.7)

±

√ 1 dz = 2i sin−1 (1/ 5) γ1 z

where the inverse sine is chosen between −π/2 and π/2. You now have three similar integrals to evaluate. Add them, divide by 2π i . . . , and do some very careful bookkeeping on the domains of the inverse trigonometric functions that occur. It can be done; in a sense it is not even difﬁcult – but it is hardly to be recommended. It is a little better (but not much) to work from the ‘continuous choice of argument’ deﬁnition. This starts you at stage (7.7) above for each subpath, with the same bookkeeping problems at the end. Here is a more civilised method. Divide γ into subpaths δ1 , δ 2 as in Figure 7.8. Now w(γ , 0) = w(δ 1, 0) + w(δ 2, 0). Since δ 1 lies entirely within the cut plane Cπ , the principal value of arg is continuous on δ1 . Hence, dotting all i’s and crossing all t’s, w(δ1 , 0) = [arg(i) − arg( −i)] /2π

= [π/2 − ( −π/2)]/2π = 21

Similarly, δ 2 lies entirely within the cut plane C0, so arg0 is continuous on δ 2. Hence

7.8 Computing the Winding Number by Eye

163

Figure 7.8 A more civilised method.

w(δ 2, 0)

= [arg0(−i) − arg0 (i)]/ 2π = [−π/2 − (−3π/2)]/2π = 21

Adding, w(γ , 0) = 1. Clearly this process can be telescoped: the arg calculations merely conﬁrm, in a predictable way, the obvious. We summarise this method as follows: (1) Break γ into convenient pieces, each lying in some cut plane. (2) For each piece, compute the contribution to the winding number as the difference between the arguments of the end points, using the relevant continous arg – that is, argα on Cα – or, geometrically, ﬁnd the angle subtended at the origin by the two end points of the subpath, with the appropriate sign. (3) Add these contributions. It helps if the dissection of γ into subpaths is performed by drawing a single straight line through the origin, because then the contributions are always ± 21 . Thus in Figure 7.9, the line L cuts the path γ into four segments AB, BC, CD, and DA.

Figure 7.9 Cutting the path along line L makes the calculation of the winding number easy.

164

Angles, Logarithms, and the Winding Number

Figure 7.10 A more complicated example where cutting the path along a suitable line makes the

calculation of the winding number easy.

Now w(γ , 0)

= w(AB, 0) + w(BC, 0) + w(CD, 0) + w(DA, 0) = 21 + 21 + 21 + 21 =2

Similarly, the path in Figure 7.10 gives w(γ , 0) = w(AB, 0) + w(BC, 0) + w(CD, 0) + w(DE, 0) + w(EF, 0) + w(FA, 0)

= 21 + 21 + 0 + (− 21 ) + 0 + 21 =1

This method is essentially the mathematical equivalent of ‘run your ﬁnger along the path and count half-turns’. The ingredients needed to make it rigorous are not complicated evaluations of args. The real crunch comes in showing that the path really does cross the chosen line L at the points A, B, C, and so on; that it crosses the line only at those points; and that it passes through these points in the stated order; and that the sense (clockwise or anticlockwise) of each subpath is as in the diagram. It is also worth bearing in mind that a smooth path may cross a line inﬁnitely many times, so this method is not foolproof. But whenever (as is always the case in the sequel) γ is speciﬁed in a straightforward way – such as by a simple formula, a polygon, or a series of straight lines and circular arcs – these ingredients are easily supplied. We then make no further fuss when we assert that a certain winding number takes a particular value. Comparison of the ﬁrst ‘bad’ method in Example 7.12 with the ﬁnal ‘good’ one gives a striking illustration of the dangers of blind ‘formula-crunching’. Complex analysis is highly geometric, and the geometry should not be despised.

7.9

Exercises 1. Compute the√ principal values of √ the arguments of the√following complex numbers: 1 + i, 21 + i 23 , (1 + i)3 , ( 21 + i 23 )243 , (1 + i) 2( 21 + i 23 )3.

165

7.9 Exercises

2. Let arg z denote the principal value of the argument of z ² = 0, that is, arg z ≤ π . For real x, y with x < 0 show: (i) limy→0 arg(x + i| y| ) = π (ii) limy→0 arg(x − i| y| ) = −π Compute the corresponding limits when x = 0 and when x > 0.

−π
0), ﬁnd all z ∈ C such that zn = reiθ (v) If zn1 = zn2, show that z1 = ωz2 , where ω is an nth root of unity. 9. For z, β ∈ C, z ² = 0, deﬁne the principal value of zβ to be zβ = exp(β Log z) Compute the principal values of the following powers: 1

2

(−2)

2

ii

(3 − 4i)1+i

2i

10. Using the notation of Exercise 9, for z ﬁxed a ∈ Cπ , what is d(az )/dz?

(3 + 4i)5

∈ Cπ , show that d(zβ )/dz = β zβ−1 . For

11. Let α ∈ R, β

∈ C. For z ∈ Cα , deﬁne (zβ )α = exp(β log α z) Compute all possible values of (ii )α , (2i )α , ((3 − 4i)1+i )α for various values of α. For ﬁxed z, show that as α varies, (zβ )α takes only ﬁnitely many values when β is a rational number. If m, n ∈ Z, n > 0, show that

((zm/ n)α )n

= zm

What is ((zn)m /n )α ? 12. Using the notation of Exercise 11, for z d((az )α ))/ dz. 13. Describe the image of the functions f : Cπ by the principal values of the following: z1/2

∈ Cα ,

compute d((zβ )α )/dz and

→ C geometrically where f (z) is given

z1/3

zi

14. Let α ∈ R, β

∈ C, z ∈ Cπ . By writing β = u + iv, ﬁnd |(zβ )α | and arg (zβ )α Show that | (zβ )α | is independent of α if and only if β is real.

For positive integers m, n let f (z) be the principal value of zm/ n. Describe f : Cπ → C geometrically.

7.9 Exercises

167

15. Express Log (1 + z) as a power series in z of the form Log (1 + z) =

∞ º n=0

(|z|

a n zn

< R)

specifying the coefﬁcients an and the radius of convergence R. 16. Show: √ (i) cos( −i Log (z + √z2 − 1)) = z (ii) sin(´ −i Log (i(z z2 − 1))) = z ´ + µµ i i+z (iii) tan Log =z 2 i−z Use these properties to express cos−1 z, sin−1 z, tan−1 z in terms of the logarithm. 17. Show that all the values of cosh −1 z have the form cosh−1 z

= log(z +

³

z2 − 1)

for all possible values of the logarithm and square root. In the same sense, show that

³

sinh−1 z = log(z + z2 + 1) µ ´ 1+z − 1 1 tanh z = 2 log 1−z

= {z ∈ C : ii −+ zz ∈ Cπ }. Describe D geometrically. Deﬁne the principal value of the inverse tangent tan−1 : D → C to be µ ´ i+z tan−1 z = 21 i Log i−z

18. Let D

(note the principal value of the logarithm). Show that tan−1 is differentiable in D with derivative 1/(1 + z2 ). 19. Draw the following paths and specify all the continuous choices of argument along them: (i) γ (t) = 2e−it (t ∈ [0, 4π ]) (ii) γ (t) = t + i(1 − t) (t ∈ [0, 1]) (iii) γ (t) = t¶− 1 + it2 (t ∈ [−1, 1]) t + i(1 − t) (t ∈ [0, 1]) (iv) γ (t) = 1 + i(t − 1) (t ∈ [1, 2]) In each case compute the winding number of the path round the origin. 20. Find the winding number w(γ , z0 ) for each of the following choices of γ , z0: (i) γ (t) = 2e−it (t ∈ [0, 2π ]); z0 = 1, 3i (ii) γ (t) = t + i(1 − t) (t ∈ [0, 1]); z0 = 1 + 1, −i, 10i

21. For each of the following, draw a picture of the path and use a sensible method to » compute γ 1/(z − z0 )dz: (i) (ii) (iii)

γ (t) = te−it (t ∈ [π , 5π ]) z0 = 0 γ (t) = −it (t ∈ [0, 1]) z0 = 1 γ (t) = it (t ∈ [−1, 1]) z0 = 1

168

Angles, Logarithms, and the Winding Number

Figure 7.11 Compute (by eye) the winding numbers of these closed paths round a point in each

component A, B, C , . . . .

γ (t) = σ + [1, 2] + ρ + [−2, −1] z0 = 0 where σ (t) = ei(π−t) , (t ∈ [0, π ]) and ρ (t) = 2e −it (t ∈ [0, 4π ]). 22. Compute (by eye) the winding number of the given closed paths round a point in each of the connected components A, B, C, . . . of the complement, drawn in Figure 7.11. (iv)

8

Cauchy’s Theorem

The Fundamental Theorem of Contour Integration, Theorem 6.33, tells us that if D is a domain and f : D → C has an antiderivative in D, then the integral of f along a path in D from z0 to z1 can be calculated using the formula

±

γ

f

= F(z1) − F(z0)

In particular, if γ is closed, so z1 = z2 , then

±

γ

f

=0 ²

Cauchy’s Theorem goes further. It states conditions under which γ f = 0 when there is no initial reason for f to have an antiderivative. There are many different versions of Cauchy’s Theorem – or, to be precise, many different theorems of this type. Cauchy was the ﬁrst to publish such a result, announcing it in 1813 and getting it into print in 1825. Gauss was aware of the basic idea in 1811, but the accolade goes to Cauchy because he was the ﬁrst to make it public. Both Gauss and Cauchy realised the basic fact that if γ does not wind² round points outside D (that is, the winding number round such points is zero) then γ f = 0. For instance, f (z) = 1/z has the single point 0 outside its domain, so if the closed contour γ ² does not wind round the origin, γ f = 0, Figure 8.1.

/

Figure 8.1 The integral of 1 z along this closed path is zero.

170

Cauchy’s Theorem

Figure 8.2 A star domain.

Our main aim in this chapter is to establish the following version of Cauchy’s Theorem: If f is differentiable in a domain D and w(γ , z) = 0 for all z ± ∈ D, then

±

γ

f

= 0.

We start the theory rolling by proving the special case where the contour is a triangle. Then we prove a theorem that requires a restriction on the domain D rather than the contour γ . we say that D is a star domain if it contains a point z∗ such that for any other ² point z ∈ D the line segment [z∗ , z] ⊆ D, Figure 8.2. We then deﬁne F(z) = [z∗ ,z] f and use the triangle version of the theorem to show that F is an antiderivative of f . This ² means that γ f = 0 for any closed contour in a star domain. In particular, a disc is a star domain, and this leads to a very signiﬁcant result. For a differentiable function f in a general domain D, we may not be able to ﬁnd an antiderivative F : D → C. But if we restrict attention to any disc ± ⊆ D, there is an antiderivative F : ± → C. Thus an antiderivative may not exist globally throughout D, but it does exist locally in any neighbourhood Nr (z0 ) ⊆ D for any z0 ∈ D. Using the Paving Lemma 2.33, any contour γ in an arbitrary domain D can be written as γ = γ1 + · · · + γn where each subcontour γr lies in a disc Dr ⊆ D. In the (star domain) Dr we can choose a step path σr with the same end points as γr and (by the ² ² existence of an antiderivative in Dr ), σr f = γr f . If we set σ = σ1 + · · · + σn , then ² ² σ f = γ f , Figure 8.3.

Figure 8.3 Reduction of a contour integral to an integral along a step path.

²

This reduces the investigation of γ f to the case of an integral along a step path which can be attacked by geometrically inspired methods.

σ,

8.1 The Cauchy Theorem for a Triangle

8.1

171

The Cauchy Theorem for a Triangle At the end of the nineteenth century, amongst many different versions of Cauchy’s Theorem, a most ingenious proof for a triangular contour was conceived by Eliakim Hastings Moore. Earlier proofs usually insisted that the function f should have a continuous derivative f ² . By restricting the contour to a triangle, Moore’s proof requires only that f ² exists throughout D. It therefore provides a suitable basis for the development of the theory for all differentiable functions. For z1, z2 , z3 ∈ C, let T (z1, z2, z3 ) be the set of points inside and on the triangle with vertices z1 , z2, z3. Formally, T(z1 , z2 , z3)

= {z ∈ C : z = λ1z1 + λ2z2 + λ3z3 , λj ∈ R, λj ≥ 0, λ1 + λ2 + λ3 = 1}

where j = 1, 2, 3. The boundary contour of the triangle, composed of the three line segments that form its sides, is

∂ T (z1, z2, z3 ) = [z1 , z2 ] + [z2 , z3 ] + [z3 , z1 ]

Whenever there is no confusion we denote the triangle by T and its boundary by ∂ T. T H E O R E M 8.1 (Cauchy’s Theorem for a Triangle). Let f be a differentiable function in ² a domain D. If the triangle T lies in D, as in Figure 8.4, then ∂ T f = 0.

²

Proof. Let | ∂ T f | = c ≥ 0. We prove that c = 0 by an indirect argument. First we subdivide T into four smaller triangles T (1) , T (2) , T (3) , T (4) by joining the midpoints of the sides as in Figure 8.5. We know that

±

∂T Therefore

f

=

4 ± ³ r= 1

∂ T (r)

´ ´± ´ ³ 4 ´± ´ ´ ´ ´ ´ f ´´ c = ´´ f ´´ ≤ ´ ∂T ∂ T (r) r =1

Figure 8.4 A triangular path in D.

f

172

Cauchy’s Theorem

Figure 8.5 Subdividing the triangular path.

so we must be able to choose r such that

´ ´± ´ c ´ ´ ´ f ´ ∂ T (r) ´ ≥ 4

(If more than one r satisﬁes this inequality, choose any of those – say the one with smallest r.) Deﬁne T 1 = T (r). Then

´± ´ ´ ´ ∂T

1

´ ´ c f ´´ ≥ and L(∂ T 1) = 12 L(∂ T) 4

Repeat this process of subdivision to get a sequence of triangles T satisfying

´± ´ ´ ´ ∂T

⊇ T1 ⊇ T2 ⊇ · · · ⊇ Tn · · ·

´ ´ f ´´ ≥ ( 14 )n c and L(∂ T1 ) = ( 21 )nL(∂ T ) n ²

(8.1)

Next we get another estimate for | ∂ Tn f | using the fact that f is differentiable. Since triangles are bounded closed sets, the intersection of the nested sequence of triangles Tn contains a point z0 ∈ D. Since f is differentiable at z0 , for any ε > 0 there exists δ > 0 such that ´ ´ ´ f (z) − f (z0) ´ ´ ´ 0 such that Nε1 (z1) ⊆ D. If |h| < ε1 , the triangle T (z∗ , z1, z1 + h) lies entirely in D, Figure 8.6. Now Theorem 8.1 gives

±

[z ∗,z1 ]

f

+

This can be written as F(z1 ) + or

±

[z1 ,z1 +h]

f

± [z 1 ,z1 +h]

f

F(z1 ) − F(z1 + h) h

±

+

[z 1 +h,z∗ ]

f

=0

− F(z1 + h) = 0 = 1h

± [z 1 ,z1 +h]

f

The proof now proceeds in the same manner as Theorem 6.44. For a constant c ∈ C,

±

[z1 ,z1 +h]

hence

F(z1) − F(z1 + h) h

By continuity of f , given ε

c dz = ch

− f (z1) =

±

f (z) − f (z1 ) dz h [z1 ,z 1 +h]

> 0 there exists δ > 0 such that |z − z1 | < δ implies | f (z) − f (z1)| < ε If z lies on the line segment [z1 , z1 + h], |h| < δ implies |z − z1 | < δ so | f (z) − f (z1)| < ε The Estimation Lemma 6.41 gives

and from (8.3), if |h|

´± ´ ´ ´ [z

´

f (z) − f (z1) ´´ ε |h| = ε dz´ ≤ h |h| 1 ,z1 +h]

< δ then

´ ´ ´ ´ F(z1 ) − F(z1 + h) ´ − f (z1 )´´ ≤ ε ´ h

(8.3)

8.3 An Example – the Logarithm

Since ε is arbitrary,

F(z1) − F(z1 + h) h→ 0 h lim

so F²

= f.

175

= f (z1 ) ²

If f is differentiable in a star domain D, then γ f = 0 for all closed contours γ in D, and the integral of f between any two points in D is independent of the choice of contour between the points. C O RO L L A RY 8.4.

Proof. This is an immediate deduction from Theorems 8.3 and 6.44.

8.3

An Example – the Logarithm In Example 6.37 we showed that the integral of 1/z from −1 to 1 depends on the path between those points. We now examine how to restrict the domain to get round this problem. Later we describe a more elegant approach. The function 1/z is differentiable in the domain C \ {0}, but this is not a star domain; moreover, there is no antiderivative on this domain. However, if we restrict the function to any star domain D ⊆ C \ {0}, the results of the previous section apply in D. It is easy to see that the cut plane Cπ is a star domain, with star centre z∗ = 1. Figure 8.7 makes this clear geometrically, and a rigorous proof is straightforward. Because 1/z is differentiable in Cπ it has an antiderivative there, the principal value of the logarithm, which satisﬁes Log z1

=

±

[1,z 1 ]

1 dz z

We exploit the fact that this integral is independent of the path, provided the path is contained in Cπ , and integrate along a specially chosen contour that is not the line segment [1, z1]. Instead we proceed as follows. Let z1 = reiθ where r > 0, −π < θ < π ,

Figure 8.7 The cut plane

Cπ is a star domain with star centre z∗ = 1.

176

Cauchy’s Theorem

/

Figure 8.8 A more convenient path to integrate 1 z in

Cπ .

and deﬁne γ = γ1 + γ2 where γ1 is the line segment [1, r] and γ2 (t) See Figure 8.8. Then ± ± 1 1 iθ Log re = dz + dz γ1 z γ2 z

±

r

= 1t dt + 1 = log r + iθ

± θ 1

reit

1

= reit (t ∈ [0, θ ]).

dt

This gives an alternative approach to the complex logarithm. In particular, it provides a far more satisfying proof of the continuity of the argument in the cut plane Cπ than the prosaic version given in Section 7.2. Namely: the function Log is differentiable in Cπ , hence continuous, so its imaginary part (which is the argument of z) is also continuous there.

8.4

Local Existence of an Antiderivative Let f be differentiable in an arbitrary domain D. We know that f may not have an antiderivative that works throughout D. But D is open, so for any z0 ∈ D there is an open disc Nr (z0) ⊆ D. A function F : Nr (z0 ) → C such that F ²(z) = f (z) for all z ∈ Nr (z0) is called a local antiderivative of f . A disc is a star domain, so Theorem 8.3 tells us that f has a local antiderivative in every disc in D. This immediately simpliﬁes integration of a differentiable function along an arbitrary contour, because we can integrate along a step path instead: 8.5. If γ is a contour from z0 to z1 in a domain D, then there exists a step path ² ² from z0 to z1 in D such that γ f = σ f for every function f that is differentiable in D.

LEMMA

σ

Proof. By the Paving Lemma 2.33, γ = γ1 + · · · + γn where each γr lies in a disc Dr ⊆ D. Let σr be a step path in Dr from the initial ² of γr to its ﬁnal point. Then ² point by Corollary 8.4, in the star domain Dr , we have γr f = σr f . Now σr = σ1 + · · · + σr is a step path from z0 to z1 in D, as in Figure 8.3, and

±

γ

f

=

n ± ³ r =1

γr

f

=

n ± ³ r =1

σr

f

=

±

σ

f

8.5 Cauchy’s Theorem

8.5

177

Cauchy’s Theorem We build up to Cauchy’s Theorem in stages. First, consider a rectangle R = {x + iy ∈ C : a ≤ x ≤ b, c ≤ y ≤ d} with boundary contour

∂ R = [z1 , z2] + [z2 , z3] + [z3 , z4] + [z4 , z1 ] where z1 = a + ic, z2 = b + ic, z3 = b + id, z4 = a + id, as in Figure 8.9. ² L E M M A 8.6. If D is a domain, f is differentiable in D, and R ⊆ D, then ∂ R f = 0. Proof. Insert the opposite diagonal contours [z1, z3 ] and [z3 , z1 ]. Use Cauchy’s Theorem for a triangle, Theorem 8.1, twice, and add. The integrals along the two diagonal paths cancel. See Figure 8.10. Now we take an arbitrary closed step path σ and insert extra line segments to make up a collection of rectangles. To do this, extend all horizontal and vertical line segments of σ to inﬁnity, breaking the plane into a ﬁnite number of rectangles, some ﬁnite, R1 , . . . , R k , and some inﬁnite, Rk+1 , . . . , Rm . (What we mean by ‘rectangle’ in

Figure 8.9 Contour deﬁned by the boundary of a rectangle.

Figure 8.10 A rectangular contour is the sum of two triangular contours.

178

Cauchy’s Theorem

Figure 8.11 Extending the edges of a step path to tile

C with ﬁnite or inﬁnite rectangles.

the inﬁnite case is clear from the ﬁgure. A ‘side at inﬁnity’ is missing.) Figure 8.11 shows an example where k = 9, m = 25. In the interior of each Rn choose a point zn , and deﬁne

νn = w(σ , zn) This is independent of the choice of zn in Rn because the interior of Rn is connected. Say that Rn is relevant if ν n ± = 0. Then Rn is relevant only when σ winds round it. In particular, all of the inﬁnite rectangles Rk+1 , . . . , Rm are irrelevant, because they lie in the inﬁnite component of the complement of σ . (In Figure 8.11 the only relevant rectangles are R3 , R5 , R 7, R8.) We now demonstrate that σ can be expressed in terms of the boundary contours of relevant rectangles, by taking νn copies of each boundary ∂ Rn when νn > 0, and −νn copies of each −∂ Rn when ν n < 0. For instance, in Figure 8.11 we take −∂ R3, ∂ R5 , ∂ R 7, ∂ R8 . Cancelling the opposite segments common to ∂ R 5, ∂ R8 and to ∂ R7 , ∂ R 8 we ﬁnish up with the step contour σ . To show this works with an arbitrary closed step path σ , it is convenient to use the notation nγ to represent n copies of γ when n ≥ 0 and −n copies of −γ when n < 0. The most straightforward procedure is to start with the set of contours A

= {−σ , ν1 ∂ R1 , . . . , νk ∂ Rk }

and show that the cancellation of opposite line segments L, −L, wherever they occur, removes them all. To prove this, suppose that when as many pairs as possible have been cancelled, There remain q ± = 0 copies of some line segment L. Then L is a side of at least one ﬁnite rectangle Rs , and by allowing q to be negative if necessary, we may suppose that L is traced in the same direction as ∂ Rs . Let R r be the rectangle on the other side of L (which may be a ﬁnite rectangle or an inﬁnite one), as in Figure 8.12. The set of closed contours B

= A ∪ {− q∂ Rs}

179

8.5 Cauchy’s Theorem

Figure 8.12 Rectangles either side of L.

then simpliﬁes to have no copies of L. If we compute the winding numbers of the contours in B round zs ∈ R s and zr ∈ Rr , then the absence of L from the simpliﬁed set of contours tells us that the two winding numbers are the same. But the winding number round zs is

ν1w( ∂ R1 , zs) + · · · + νk w(∂ Rk , zs) − w(σ , zs) − qw(∂ Rs , zr) = −q and around zr it is

ν1 w(∂ R1, zr ) + · · · + νk w(∂ Rk , zr ) − w(σ , zr ) − qw(∂ Rs, zr ) = 0

Hence q = 0 as required. This conﬁrms that σ may be obtained by taking νn copies of each relevant contour ∂ Rn and deleting opposite line segments wherever they occur. 8.7. Let σ be a closed step path in a domain D such ² that w(σ , z) D. Then, for any function f differentiable in D, we have σ f = 0.

LEMMA

z ±∈

= 0 for all

Proof. We express σ in terms of relevant rectangles, as above, and show that every relevant rectangle R n must lie inside D. By deﬁnition, if z lies in the interior of Rn then w(σ , z) = νn ± = 0 so z ∈ D. On the other hand, a point z on the boundary ∂ R n either lies on σ (and hence in D) or it is in the same component of the complement of σ as points in the interior of R n, whence w(σ , z) = νn ± = 0 and again z ∈ D. By cancelling contributions along opposite contours,

±

σ

f

=

k ³ n=1

νn

±

∂ Rn

f

Integrals on the right need be considered only when νn so lies inside D. Then, by Lemma 8.6,

±

²

and σ f

= 0.

∂ Rn

f

±= 0, in which case Rn is relevant

=0

We now reach the focal point of this chapter: Let f be differentiable in a domain D and let γ be a closed contour in ²D that does not wind round any points outside D (that is, w(γ , z) = 0 when z ± ∈ D). Then γ f = 0. T H E O R E M 8.8 (Cauchy’s Theorem).

180

Cauchy’s Theorem

²

²

Proof. By Lemma 8.5 there exists a step path σ in D such that σ φ = γ ² ² function φ that is differentiable in D. In particular, σ f = γ f . For z0 ± ∈ D the function φ(z) = 1/(z − z0) is also differentiable in D, so w(σ , z0) =

²

By Lemma 8.7, σ f

8.6

±

1

φ = 21π i

2π i σ

= 0. Hence

²

=

γf

²

±

γ

φ for any

φ = w(γ , z0 ) = 0

= 0.

σf

Applications of Cauchy’s Theorem The version of Cauchy’s Theorem that we have just proved has far wider applications than simply showing that integrals round certain closed contours must be zero. It lets us calculate non-zero integrals as well. For example, suppose that γ1 and γ2 have the same winding number round all points outside D, so that w(γ1, z) = w(γ2, z) when z ± ∈ D. Let z1 be the point where γ1 begins and ends, and let z2 be the point where γ2 begins and ends. Take any contour σ from z1 to z2 in D, Figure 8.13.

Figure 8.13 Creating a closed path from two paths.

Let γ = γ1 + σ − γ2 − σ . This is a closed contour in D, and w(γ , z)² = 0 for z ± ∈ D because the winding number is additive. By Cauchy’s Theorem, γ f = 0. Therefore

±

γ1 and

f

+

±

f

σ

± ²

γ1

f

±

γ2

=

f

±

σ

f

=0

±

γ2

f

If we wish to compute γ f , we may be able to ﬁnd another contour γ2 as above, for 1 ² which γ2 f is simpler. The technique of introducing opposite contours σ , −σ whose contributions eventually cancel is also very useful. In particular, it lets us prove a much more powerful Cauchytype theorem:

181

8.6 Applications of Cauchy’s Theorem

Figure 8.14 Joining a set of closed contours to form a single closed contour.

8.9 (Generalised Version of Cauchy’s Theorem). Suppose that γ1 , . . . , γn are closed contours in a domain D such that THEOREM

w(γ1, z) + · · · + w(γn , z) = 0 for all z and let f be differentiable in D. Then

±

γ1

f

+ ··· +

±

γn

f

±∈ D

=0

Proof. Suppose that γr begins and ends at zr (1 ≤ r ≤ n). Choose any z0 contours σ 1, . . . , σn in D that join z0 to z1, . . . , zn respectively, Figure 8.14. Then

∈ D and

γ = σ1 + γ1 − σ1 + · · · + σn + γn − σn

is a closed contour beginning and ending at z0, and w(γ , z) = 0

for all z

±∈ D

²

(again, by additivity of the winding number). By Cauchy’s Theorem, γ f n ± ³ r=1

Therefore

σr

f

+

±

γr

n ± ³ r =1 γr

8.6.1

±

f

f

=0

σr

f

= 0, so

=0

Cuts and Jordan Contours Given two closed contours γ1 , γ2 in D and a contour σ in D from a point on γ1 to a point on γ2 , we call the pair of contours σ , −σ a cut from γ1 to γ2. There is a historical reason for this name. Earlier versions of Cauchy’s Theorem were invariably proved for Jordan contours. A closed Jordan contour is a closed contour γ : [a, b] → C that does not cross itself. That is, a ≤ t1

< t 2 < b implies γ (t1) ±= γ (t 2)

182

Cauchy’s Theorem

It is intuitively obvious, but difﬁcult to prove analytically, that every closed Jordan contour γ separates the plane into two components, the points O(γ ) outside γ and the points I(γ ) inside γ , and that these are both connected sets, Figure 8.15. ² Early versions of Cauchy’s Theorem state that if γ and I(γ ) lie in D, then γ f = 0. In applications it was then necessary to introduce ‘cuts’ to manufacture Jordan contours. For instance, suppose that f is differentiable everywhere except at z0 and two Jordan contours γ1, γ2 both wind once round z0 as in Figure 8.16 (left). Two cuts are made in Figure 8.16 (right) to create two new Jordan contours so that f is differentiable inside each of them. The integral round each new ² Jordan ² contour is then zero, and cancelling the contributions from the cuts we get γ f = γ f . Such methods usually relied on 1

2

geometric intuition, sometime unsupported by analytic proof. By introducing the winding number to link analysis to geometry, such pitfalls can be avoided, and there is no longer any necessity to restrict the theory to Jordan contours. Instead we can deﬁne the inside of any closed contour to be I(γ ) = {z ∈ C : w(γ , z) ± = 0} and the outside to be O(γ ) = {z ∈ C : w(γ , z)

= 0}

Figure 8.15 Closed Jordan contour and its inside and outside.

Figure 8.16 Left: Two Jordan contours. Right: ‘Rewiring’ the contours using cuts.

183

8.7 Simply Connected Domains

Figure 8.17 Inside and outside components of the complement of a closed contour.

In general, neither I(γ ) nor O(γ ) need be connected. For example, in Figure 8.17, O( γ ) has two components O1 and O2 , and I(γ ) has three, I1 , I2 , and I3 . The Jordan Contour Theorem (which we do not prove, but see Exercise 9 below for the step path case) then says that the outside and inside of a closed Jordan contour are both connected. For an arbitrary closed contour we can rephrase Cauchy’s Theorem as stated in Theorem 8.8 to get: 8.10. Let f be differentiable in a domain D. If a closed contour ² inside I(γ ) lies in D, then γ f = 0. THEOREM

8.7

γ

and its

Simply Connected Domains Theorem 6.44 states that if D is a domain and f : D → C is differentiable in D, then ² f = 0 for all closed contours γ in D if and only if f has an antiderivative in D. We γ can now state the precise conditions under which this happens for all functions f that are differentiable in D. 8.11. A domain D is simply connected if w(γ , z) contour γ in D and every z ± ∈ D. Equivalently, I(γ ) ⊆ D for every closed contour γ in D. D E FI N I T I O N

= 0 for every closed

Intuitively, a domain is simply connected if it has no holes. We now have:

²

8.12. If D is a domain, then γ f only if D is simply connected. THEOREM

= 0 for all closed contours γ in D, if and ²

Proof. If D is simply connected then Cauchy’s Theorem implies that γ f = 0 for any closed contour γ in D and any f : D → C that is differentiable in D. Conversely, if D is not simply connected, there exists a closed contour γ0 in D and z0 ∈ D such that

184

Cauchy’s Theorem

w(γ0, z0) ± = 0. Let φ(z) ² γ φ = w(γ0, z0) ±= 0.

8.8

= 1/(2π i(z − z0)). Then φ : D → C is differentiable in D and

Exercises 1. State which of the following are star domains, proving the existence of a star centre for those that are, and justifying your answer for those that are not.

{z ∈ C : z ±= x + 0i where |x| ≥ 1} {z ∈ C : |z| > 1} {z ∈ C : z ±= eit for t ∈ [0, π ]} {z ∈ C : |z| > 1 and either im z > 0 or re z > 0} Let D = C \ { 0}. For z0 ∈ D, specify a local antiderivative in some neighbourhood

(i) (ii) (iii) (iv) 2.

of z0 for each of the following functions:

(i) (ii) (iii) (iv) (v)

1/z 1/z2 (z + 1)/ z2 (cos z)/z (sin z)/ z

3. Let

γ1(t) = −1 + 21 eit (t ∈ [0, 2π ]) γ2(t) = 1 + 12 eit (t ∈ [0, 2π ]) γ (t) = 2e−it (t ∈ [0, 2π ])

If f (z) = 1/(z2 − 1) use Theorem 8.9 to deduce that

±

γ

f

=

±

γ1

f

+

±

γ2

f

Interpret this statement in terms of the winding numbers of γ , γ1, γ2 round 1, −1. 4. Show that D

= {z ∈ C : z ±= ³1} is not simply connected. Let L 1 = {x + iy ∈ C : y = 0, x ≤ −1} L 2 = {x + iy ∈ C : y = 0, x ≥ 1} D0 = D \ { L1 ∪ L 2)

Show that D0 is simply connected. Is it a star domain? Does f (z) = 1/(z2 − 1) have an antiderivative in D0? In each case justify your answer. 5. Prove that every quadrilateral whose edges do not cross is a star domain. What about pentagons? Hexagons?

6. Let D = {²z ∈ C : z ± = ³i } and let γ be a closed contour in D. Find all the possible values of γ 1/(z2 + 1)dz. If σ is a contour from 0 to 1, ﬁnd all possible values of

²

σ 1/(z

2

+ 1)dz.

8.8 Exercises

185

7. Let γ1 = S1 + L − S2 − L, γ2 = S1 + L + S2 − L, where S1 (t) = eit (t ∈ [0, 2π ]) S2 (t) = 2eit (t L = [1, 2]

∈ [0, 2π ])

Describe the inside and outside of γ1 and γ2. Let f (z) = (cos z)/z, By writing cos z as a power series and considering f (z) = ² ² (1/ z) + g(z), or otherwise, compute γ1 f and γ2 f . Compare the results with Theorem 8.10.

8. Let D = {z ∈ C : z ± = z1 , . . . , z ± = zk } where zj ∈ C, and suppose that f is differentiable in D. Show that for any closed contour in D,

±

γ

f

=

k ³ r −1

± nr

Sr

f

where Sr is a sufﬁciently small circle centre zr and nr is an integer. If limz →z r (z − zr )f (z) = ar ∈ C for r = 1, . . . , k, show that

±

γ

f

=

k ³ r−1

2π inr ar

9. Jordan Contour Theorem for Step Paths. Deﬁne a crossing of a closed step path σ : [a, b] → D to be a point σ (t1 ) in the image of σ , such that

σ (t1 ) = σ (t2 )

(a ≤ t1

< t2 < b)

Say that σ is simple if it has no crossings. Prove the Jordan Contour Theorem for step paths: Let σ be a simple closed step path in C. Then (i)

C \ σˆ has precisely two connected components. One (call it I(σ )) is bounded,

the other (call it O(σ )) is unbounded. (ii) Either w(σ , z) = 1 for all z ∈ I(σ ), or w(σ , z) = −1 for all z ∈ I(σ ). (iii) If z ∈ O(σ ), then w(σ , z) = 0.

10. Cauchy’s Theorem for a Simple Closed Step Path. Prove Cauchy’s Theorem for a² simple closed step path σ : If σ does not wind round any point not in D, then σ f = 0. Hint: One way to do this is as follows: (1) Construct a set of rectangles R r as in Section 8.4. (2) Prove that a rectangle Rr is relevant if and only if it lies in I(σ ). (3) Prove that I(σ ) ⊆ D, observing that by deﬁnition I(σ ) winds around any point in I(σ ). ² (4) Prove that Rr f = 0, by observing that Rr is a star domain. (5) Deﬁne a closed path τr round the boundary of each rectangle, having the same winding number as σ round any point inside the rectangle. (This is either always 1 or always −1 by Exercise 9 (ii).)

186

Cauchy’s Theorem

(6) Show that

³± r

Rr

f

=

±

σ

f

because edges of the Rr that are not contained in the image of σ cancel is pairs. (7) Deduce Cauchy’s Theorem for σ .

11. Suppose that σ1, σ2 are simple closed step paths in contained in I(σ2 ). Prove that I(σ1 ) ⊆ I(σ2 ).

C, and that the image of σ1 is

9

Homotopy Versions of Cauchy’s Theorem

The results in this chapter are not essential for any later parts of the text, except for Chapter 16.

Cauchy’s Theorem is one of the foundation stones of ±complex analysis. It resolves two apparently conﬂicting features of contour integrals γ f when f is ﬁxed but γ can change. On the one hand, γ can be altered in fairly drastic ways with no effect on the integral – for instance, replacing a general path by a step path. On the other hand, changing a semicircular path in the upper half-plane of C into a semicircular path in the lower half-plane of C completely changes the integral of 1/z between −1 and 1, as in Section 6.10. The version of Cauchy’s Theorem in Theorem 8.8 explains these features: what really matters is the winding number of γ around points that lie outside the domain D of f . This result emphasises the topology of the domain, and how the path lies within it. To improve our understanding, we examine these topological issues in more detail. In fact, we do so in two ways. In this chapter we consider continuous deformations of γ , captured by the topological notion of homotopy. In Chapter 16 we discuss a related concept, homology. Both concepts formalise the idea that the domain D has ‘holes’, and the integral depends on how the path wanders around the domain, relative to these holes. However, they do this in two different, though related, ways. Homotopy is easier to visualise and geometrically quite natural. Homology is algebraically simpler, once some initial difﬁculties have been overcome, but it can appear artiﬁcial and contrived. We reformulate Cauchy’s Theorem from these two viewpoints. Each offers fresh insights. We also show that an arbitrary path γ in a domain D can be replaced by a suitable approximating polygon without changing the integral of any function that is differentiable in D. This generalises Cauchy’s Theorem so that it applies to any closed path, removing the ‘piecewise smooth’ condition required of a closed contour.

9.1

Informal Description of Homotopy Homotopy is one of the central ideas in topology. It describes topological features of spaces in terms of families of continuously varying paths. For example, consider the

188

Homotopy Versions of Cauchy’s Theorem

γ

γ

Figure 9.1 Deforming the semicircular path 0 continuously into 1 .

anticlockwise semicircle γ0 in C from 1 to −1 deﬁned by

γ0(t) = eit

∈ [0, π ]) and the clockwise semicircle γ1 in C from 1 to −1 deﬁned by γ1 (t) = e−it (t ∈ [0, π ]) Figure 9.1 shows visually that we can deform γ0 into γ1 by displacing it vertically. In formulas, let

(t

γs (t) = (1 − s)eit + se−it

Then γ0 and γ1 are the two semicircular paths, and the path γs varies continuously as s varies continuously from 0 to 1. However, if the paths are not allowed to pass through the origin ( for example, they may be paths in the domain of a function that has a singularity at the origin, such as 1 /z), this continuous deformation is not permitted, because γ1/ 2 passes through the origin at t = π/2. A little experiment strongly suggests that no continuous deformation from γ0 to γ1 exists if no intermediate path meets the origin. The origin is an obstacle to deforming the path, and any attempt to do so causes the path to get ‘hung up’ at the origin. The origin creates a hole, and the path cannot cross the hole. This suggestion is in fact true, but a proof requires some care: it can, for example, be based on properties of the winding number. Indeed, the winding number is a simple example of a homotopy invariant: a way to distinguish topologically different spaces using notions of homotopy. In this chapter we apply homotopy ideas to complex integration by considering what ± happens to γ f when we allow the contour γ to vary continuously. We derive precise conditions under which γ can be deformed continuously without changing the integral. This can be done in two ways. One is to ﬁx the end points z0, z1 and allow the contour between them to be continuously deformed. The other is to perform a continuous deformation of a closed contour. In both cases the deformations must keep the path inside a domain D on which f is differentiable. Figure 9.2 illustrates these two possibilities.

9.2 Integration Along Arbitrary Paths

189

Figure 9.2 Two types of continuous deformation of a contour. Left: Keeping end points z1 , z 2 of

ﬁxed. Right: Deforming a closed contour γ .

γ

Both of these methods are special cases of a single result, The Cauchy Theorem for a boundary, which is proved in Section 9.3. Before this, in Section 9.2 we show how ± the conditions on γ can be relaxed when f is differentiable to deﬁne γ f along any path (recall that a contour is piecewise smooth). This lets us vary γ freely without having to worry whether the intermediate paths involved are contours.

9.2

Integration Along Arbitrary Paths Suppose that f : D → C is differentiable in the domain D. Up to now we have usually integrated such a function along a contour in D – a piecewise smooth path. However, because differentiable functions, are fairly well behaved, the precise path taken turns out not to be so important. We saw this in Lemma 8.5, where we were able to replace a contour by a step path and get the same result. We can use the same technique to deﬁne an integral along an arbitrary path γ : [a, b] → D. This deﬁnition agrees with the Riemann integral if that happens to be deﬁned – an application of Cauchy’s Theorem that we leave to the interested reader. The Paving Lemma gives a partition a = t0 < t1 < · · · < t n = b such that each subpath γr deﬁned on [tr −1, t r ] lies in a disc Dr ⊆ D. Let λ be the approximating polygon λ = λ1 +· · ·+λn , where λr is the line segment from γ (tr−1 ) to γ (tr ). Then λr lies inside Dr for all r, so λ lies inside D, and we can deﬁne the integral of f along γ to be

²

γ

f

=

²

λ

f

See Figure 9.3. This deﬁnition can be proved to be independent of the choice of the partition that determines λ, provided these are chosen using the Paving Lemma as described above. For if extra division points are introduced between t r−1 and t r , say

< s1 < · · · < sk < tr then γ (s1 ) , . . . , γ (sk ) ∈ D so the integral of f along λr has the same value as that along the polygon with vertices γ (tr −1 ), γ (s1 ) , . . . , γ (sk ), γ (tr ), by Theorems 6.33 and 8.3. tr −1

See Figure 9.4.

190

Homotopy Versions of Cauchy’s Theorem

Figure 9.3 Replacing a continuous path by a polygonal one.

Figure 9.4 Subdividing a polygonal path inside a disc in D.

Thus a ﬁner partition of [a, b] does not change the integral. Given any two polygonal approximations λ, µ obtained using the Paving Lemma, let ν be the polygon whose ± ± ± vertices are those of λ, µ taken together. Then λ f = ν f = µ f , so the deﬁnition of the integral is independent of the polygonal approximation – provided that is chosen using the Paving Lemma. An arbitrary polygonal approximation to γ will not do. In Figure 9.5, suppose that D = C \ {z0}. The polygon λ with vertices z1, z2 , z3 , z4 , z5, z6, z7 is an approximation to γ found using the± Paving with vertices z1, z6, z7 is not. If ± Lemma, but ±the polygon ± f = 1/ (z − z0 ) then λ f − µ f = 2π i, so λ f ± = µ f . It is straightforward to check that Theorems 8.8, 8.9, 8.10, and 8.12 of Chapter 8 also hold for arbitrary paths, however wild – even ‘space-ﬁlling’ curves, Section 2.9. For instance, if f is differentiable in D and does not wind round points outside D, then ± γ f = 0. With such results at our disposal we can widen the scope of our ideas and introduce general notions from topology. These provide further insight into the various versions of Cauchy’s Theorem and how they relate to each other. We return to the topological theme and develop it further in Chapter 16, using results from Section 9.3.

9.3 The Cauchy Theorem for a Boundary

191

Figure 9.5 A different polygon can lead to a different value for the integral.

Figure 9.6 A rectangle and the path formed by its perimeter.

9.3

The Cauchy Theorem for a Boundary The formal deﬁnition of homotopy (Deﬁnition 9.7 below) uses a map from a rectangle into C. The top and bottom edges of the rectangle determine two paths, and parallel segments of its interior determine a continuous deformation of one path into the other. We therefore begin by considering maps deﬁned on rectangles, and properties of their boundaries. Let R be the rectangle R = {x + iy ∈ C : a ≤ x ≤ b, c ≤ y ≤ d} with boundary contour ∂ R, Figure 9.6. (Here a ≤ b and c ≤ d are real.) Suppose that ∂ R : [0, p] → C is parametrised by arc length, measured anticlockwise, where p = 2(b − a) + 2(d − c) is the perimeter of R, Figure 9.7. Given a continuous map φ : R → C, we deﬁne the boundary of φ to be

∂φ (t) = φ(∂ R(t))

(t ∈ [0, p])

192

Homotopy Versions of Cauchy’s Theorem

Figure 9.7 Parametrising the perimeter by arc length.

Figure 9.8 Boundary of the map

φ in Example 9.2.

φ + iy) = xeiπ y (a ≤ x ≤ b, c ≤ y ≤ d) then ∂φ = ±1 +±2 +±3 +±4

Example 9.1. If (x as in Figure 9.8.

9.2. Here we are using notation in a ﬂexible way by writing ±j for the image of ±j . Technically this is sloppy, but it avoids undue proliferation of ‘image’ symbols. Once we understand that a path differs from its image, we have earned the right to be sloppy. We exert that right repeatedly. REMARK

={ + = +

≤ ≤ ≤ ≤ } + ∈ + ≥ + ∈ + < φ + + = φ ∂φ

Example 9.3. Let R x iy : 0 x 2, 0 y 2 . For x iy R and x y 1, deﬁne (x iy) x iy, and for x iy R and x y 1, deﬁne (x iy) to be the reﬂection of x iy in the line x y 1. Then the effect of is to fold over the bottom left-hand corner of R as in Figure 9.9. This shows that the image of need not be the boundary of the image (R).

φ +

+

φ

However, Figure 9.9 provides the clue that all points inside ∂φ lie in the image φ(R). We prove this is true in general: LEMMA

9.4. If φ : R

→ C is continuous, then I (∂φ) ⊆ φ(R).

9.3 The Cauchy Theorem for a Boundary

Figure 9.9 Boundary of the map

193

φ in Example 9.3.

Proof. Suppose to the contrary that there exists z0 ∈ I(∂φ) but z0 ± ∈ φ (R). Let D = C \ {z0}. Then D is a domain and φ (R) ⊆ D. If f (z) = 1/(2π i(z − z0 )) then f is differentiable in D and ²

∂φ

f

= w( ∂φ, z0)

is a non-zero integer. Subdivide the rectangle R by its vertical and horizontal bisectors into four equal rectangles R(1) , R(2) , R(3) , R(4) and let φ(r) be the restriction of φ to R(r) . Then for r = 1, 2, 3, 4 the boundaries ∂φ (r) are all closed paths in D. Because

²

∂φ

f

=

4 ² ³

(r) r =1 ∂φ

f

and

²

∂φ (r)

f

= w(∂φ (r), z0)

at least one of the four integrals is a non-zero integer. Denote the corresponding rectangle R(r) by R1 and the restriction of φ to R 1 by φ1. Similarly dividing R1 into four equal rectangles and repeating the process yields a nested sequence of rectangles

±

R

⊇ R1 ⊇ · · · ⊇ Rn ⊇ · · ·

where each ∂φn f is a non-zero integer. The sequence of rectangles contains a (unique) point z1 ∈ R. For ε have Nε (φ(z1 ) ⊆ D. By continuity of φ there exists δ > 0 such that

= |φ (z1) − z0|, we

φ(Nδ (φ(z1 ) ∩ R) ⊆ N ε (φ (z1)) For suitably large N we have R N ⊆ Nδ (φ (z1)), so ∂φN is a closed path in the disc ± Nε (φ(z1 )). But f is differentiable in Nε (φ (z1 )), a star domain,. So ∂φ f = 0 by ± Corollary 8.4, contradicting ∂φn f being a non-zero integer.

N

We can now prove an important variant of Cauchy’s Theorem. But ﬁrst (having decided not to rely on the topological notion of compactness) we need a twodimensional analogue of the Paving Lemma. 9.5. Let R be a rectangle and let φ : R → D be a continuous map, ⊆ C a domain. Let R be divided into n2 smaller rectangles, by subdividing its

P RO P O S I T I O N

with D

194

Homotopy Versions of Cauchy’s Theorem

sides into n equal parts. Then for sufﬁciently large n, the image of every subrectangle is contained in some disc in D. Proof. The proof uses the same ideas as that of Lemma 2.33. Say that a rectangle is pavable if if it can be bisected repeatedly a ﬁnite number of times into closed subrectangles, so that the image of every subrectangle obtained in this manner lies inside a disc in D. Otherwise, the interval is unpavable. Suppose the result is false. Then R is unpavable. Split it into four closed subrectangles by bisecting the sides. At least one these subrectangles, call it R 1, must be unpavable. Bisect R 1 to obtain an unpavable rectangle R2 , and so on. This sequence continues indeﬁnitely because we are assuming the result false, so we get a nested sequence of rectangles R ⊇ R1

⊇ R2 ⊇ · · ·

each half the size of the previous one. The intersection of these rectangles is a unique point z0 ∈ R. However, D is open so there is a disc N ε (φ(z0 )) ⊆ D. If k is large enough, Rk ⊆ Nε (φ(z0 )), a contradiction. Now take n = 2k . 9.6 (Cauchy’s Theorem for a Boundary). If φ : R → D is a continuous map from a rectangle R into a domain D, as in Figure 9.10, and f is differentiable in D, then THEOREM

²

∂φ

=0

f

Proof. Cut R up into smaller rectangles Rpq by dividing it into n equal parts horizontally and vertically, where 1 ≤ p ≤ n, 1 ≤ q ≤ n. By Proposition 9.5 this can be done so that each φ(R pq) is contained in a disc Dpq ⊆ D. See Figure 9.11. Let φpq be the restriction of φ to Rpq . Then its boundary ∂φpq is a closed path in the disc Dpq ⊆ D, so

²

∂φpq

Figure 9.10 Effect of map

φ on rectangle R.

f

=0

9.4 Formal Deﬁnition of Homotopy

Figure 9.11 Subdividing the rectangle R, and the effect of

Adding up all the integrals for 1 ≤ p ≤ n, 1 opposite paths inside φ(R), we are left with

²

∂φ 9.4

f

195

φ.

≤ q ≤ m, and cancelling integrals along

=0

Formal Deﬁnition of Homotopy A ‘homotopy’ between two paths γ0 : [a, b] → D and γ1 : [a, b] → D is, roughly speaking, a continuously varying family of paths γs : [a, b] → D, where s runs over the interval [0, 1]. At the start, s = 0 and γs = γ0 ; at the end, s = 1 and γs = γ1. How do we make this precise? Notice that the whole family depends on two variables: the parameter, s, and the original variable t giving the position on the path. We can combine everything into a single function γ of (t, s) if we set

γ (t, s) = γs(t) (t ∈ [a, b], s ∈ [0, 1]) It is then natural to insist that γ , as a function of two real variables, should be continuous. We are thus led to: D E FI N I T I O N 9.7.

A homotopy in D between γ0 and γ1 , as above, is a continuous map

γ

: [a, b] × [0, 1] → D

such that

γ (t, 0) =γ0(t) γ (t, 1) =γ1(t)

for all t ∈ [a, b]

for all t ∈ [a, b]

We illustrate this idea in Figure 9.12. Note that we do not (yet) assume that γ0 and γ1 have the same start or end points. It will prove convenient to think of [a, b] × [0, 1] as a subset of C, by identifying (t, s) with t + is. Figure 9.12 captures well the continuous variation of γs , but it is misleadingly nice in that γ is one-one. There is no reason to require this in general, so a perfectly reasonable homotopy could resemble Figure 9.13.

196

Homotopy Versions of Cauchy’s Theorem

γ

γ

Figure 9.12 A homotopy between two paths 0 and 1.

γ

γ

Figure 9.13 A homotopy that is not one-one between two paths 0 and 1.

Geometrically, of course, [a, b] × [0, 1] is a rectangle, and its boundary is a closed path, with corners. If we parametrise the four edges of the rectangle in some way, and then use γ to map the result into D, we obtain paths in D that join up to give a closed path, as for example in Figure 9.12. Our aim is to apply Cauchy’s Theorem for a boundary to this set of paths. Now, there is a problem: the two edges that give γ0 and γ1 are obviously useful, but the other two edges, marked ρ , τ in Figure 9.12, are going to be a nuisance. We therefore add conditions that eliminate them. There are two obvious ways to do this: (a) Insist that each of ρ and τ goes to a single point in D. (b) Insist that ρ and τ cancel each other out. These two conditions yield two more restricted types of homotopy: ﬁxed end point homotopy, and closed path homotopy. We describe these in detail in the next two sections.

9.5 Fixed End Point Homotopy

9.5

197

Fixed End Point Homotopy Consider the rectangle R = {t + is

∈ C : t ∈ [a, b], s ∈ [0, 1]}

D E FI N I T I O N 9.8. Two paths γ0 : [a, b] → D and γ1 : [a, b] homotopic in D if there is a continuous map

→ D are ﬁxed end point

φ :R→D and points z0, z1 ∈ C such that

φ(t, 0) = γ0(t) φ(t, 1) = γ1(t) φ(s, 0) = z0 φ(s, 1) = z1

for all t ∈ [a, b] for all t ∈ [a, b] for all s ∈ [0, 1] for all s ∈ [0, 1]

as in Figure 9.14. If we let γs (t) = φ(t, s) then γs is a path in D from z0 to z1 for all s ∈ [0, 1], and as s increases continuously from 0 to 1, the path γs ‘deforms continuously’ from γ0 to γ1.

= { ∈ C : |z| < 2}, γ0(t) = t (t ∈ [−1, 1]), γ1 (t) = ∈− γ0 and γ1 are ﬁxed end point homotopic in D, where

Example 9.9. Let D z 1 i(t 1) e2 (t [ 1, 1]). Then

π −

φ(t, s) = (1 − s)γ0 (t) + sγ1(t)

Figure 9.14 A ﬁxed end point homotopy.

(t

∈ [−1, 1], s ∈ [0, 1])

198

Homotopy Versions of Cauchy’s Theorem

γ

γ

Figure 9.15 A ﬁxed end point homotopy between 0 and 1 .

As a corollary of Theorem 9.6 we deduce: 9.10. If f is differentiable in a domain D and γ0 is ﬁxed end point homotopic ± ± in D to γ1, then γ0 f = γ1 f .

THEOREM

Proof. We have a continuous map φ : R → D where

φ(t, 0) = γ0(t) φ(t, 1) = γ1(t) φ (0, s) = z0 φ (1, s) = z1

for all t ∈ [a, b] for all t ∈ [a, b] for all s ∈ [0, 1] for all s ∈ [0, 1]

as in Figure 9.15. If pr : [0, 1] → D is the path pr (t) = zr for r = 0, 1, whose image is a single point ± {zr}, then pr f = 0 and ∂φ = γ0 + p1 − γ1 + p0 . By Cauchy’s Theorem for a boundary,

²

so

∂φ

f

=

²

γ0

f

+

²

²

γ0 9.6

p1

f

f

=

²

γ1

f

+

²

p0

f

=0

²

γ1

f

Closed Path Homotopy Once more we consider the rectangle R = {t + is

∈ C : t ∈ [a, b], s ∈ [0, 1]} but we impose different conditions on φ : D E FI N I T I O N 9.11. Two (closed) paths γ0 : [a, b] → D and γ1 homotopic via closed paths in D if there is a continuous map

φ :R→D

: [a, b]

D are

9.6 Closed Path Homotopy

199

such that

φ(t, 0) = γ0 (t) φ(t, 1) = γ1 (t) φ (a, s) = φ (b, s)

∈ [a, b] for all t ∈ [a, b] for all s ∈ [0, 1] for all t

as in Figure 9.16.

Figure 9.16 Homotopy via closed paths.

Again, if we deﬁne γs (t) = φ(t, s) then γs is a closed path in D for all s ∈ [0, 1], and as s increases continuously from 0 to 1, the path γs ‘deforms continuously’ from γ0 to γ1.

| |< = { ∈ | | < ±= } | | < ρ < K, let ∈ π < ε < − | | γ = + ε ∈ [0, 2π ]), γ γ φ (t, s) = (1 − s)γ0(t) + sγ1 (t) (t ∈ [0, 2π ], s ∈ [0, 1])

Example 9.12. For z0 K, let D z C: z K, z z0 . For z0 it e (t [0, 2 ]), and for 0 K z0 let 1 (t) z0 eit (t 0(t) as in Figure 9.17. Then 0 is homotopic to 1 via closed paths in D, where

γ

Figure 9.17 Homotopy between the closed paths in Example 9.12.

200

Homotopy Versions of Cauchy’s Theorem

9.13. If f is differentiable in a domain D and closed paths γ0 , γ1 in D are ± ± homotopic via closed paths in D, then γ0 f = γ1 f .

THEOREM

Proof. We have a continuous map φ : R → D such that

φ(t, 0) = γ0(t) for all t ∈ [a, b] φ(t, 1) = γ1(t) for all t ∈ [a, b] φ (a, s) = φ (b, s) for all s ∈ [0, 1] Let σ (s) = φ(a, s) (s ∈ [0, 1]). Then ∂φ = γ0 + σ − γ1 − σ (so σ is a cut from γ0 to γ1 in the sense of Section 8.7). By Cauchy’s Theorem for a boundary,

²

∂φ

f

so

=

²

γ0

f

+

²

σ

²

γ0

f

f

=

²

γ1

f

²

σ

f

=0

²

γ1

f

A closed path γ in D is homotopic to zero or null-homotopic if it is homotopic via closed paths in D to β : [a, b] → D where β(t) = z0 for all t ∈ [a, b], with z0 ∈ D. We immediately deduce: 9.14. Let f be differentiable in a domain D and let γ be a closed path in ± D that is homotopic to zero. Then γ f = 0.

C O RO L L A RY

The geometric signiﬁcance of being homotopic to zero is that γ can be continuously deformed to a single point – more precisely, a path whose image is a single point – as in Figure 9.18. We can use a homotopy to prove that the interval of a differentiable complex function is very well behaved under reparametrisation, of the path of integration. Here we can work with merely continuous paths and parameter changes. For simplicity we do not change the parametric interval. It is a useful exercise to include that possibility (introduce an afﬁne map [a, b] → [c, d] and examine how that affects the integral). In the smooth case it is also possible to use the formula for a change of variable in an integral.

Figure 9.18 A homotopy to zero.

9.7 Converse to Cauchy’s Theorem

201

P RO P O S I T I O N 9.15. Let f be differentiable on a domain D, and let λ : [a, b] → C be ± ± a path in D. Let ρ : [a, b] → [a, b] be continuous, and let γ = λ ◦ ρ . Then γ f = λ f .

Proof. Deﬁne a homotopy σ : [a, b] × [0, 1] → D by Clearly σs (t) ∈ D for t

σ (t, s) = λ((1 − s)t + sρ(t))

∈ [a, b]. It is easy to check that σ (t, 0) = γ σ (t, 1) = λ Therefore σ is a ﬁxed end point homotopy from γ to λ . Now Theorem 9.10 applies. 9.7

Converse to Cauchy’s Theorem Starting with Cauchy’s Theorem for a boundary we have deduced ﬁrst the homotopyinvariance of the integral for ﬁxed end point or closed path homotopies, and then that the integral is zero for a path homotopic to zero. However, we can also start with Corollary 9.14 and argue the other way if we wish. First we need a simple result about changing the parameter of a path: P RO P O S I T I O N 9.16. Let γ : [a, b] → D be a path and let ρ : [a, b] → [a, b] be a continuous map such that ρ (a) = a, ρ (b) = b. Then γ is homotopic to γ ◦ ρ .

Proof. Deﬁne φ : [a, b] × [0, 1] → D by

φ(t, s) = γ ((1 − s)t + sρ(t)) This is a homotopy, and φ(t, 0) = γ (t), φ (t, 1) = γ (ρ (t)). Note that the reparametrisation ρ is not required to be a bijection here. Also: all paths γs have the same image. In effect we are performing a homotopy on the parameter. We can now prove:

A closed path γ in D is a boundary ∂φ, up to reparametrisation, is homotopic to zero.

P RO P O S I T I O N 9.17.

if and only if γ

Proof. The need to reparametrise the interval on which γ is deﬁned arises because of the way we have chosen a speciﬁc parametrisation for a boundary ∂φ . By Proposition 9.16 we can adjust the parameter provided the images of γ and ∂φ coincide. This reduces the proof to a geometric argument. We give the essence of the proof in a series of pictures, leaving the reader the (routine) task of analytic deﬁnitions and veriﬁcations required to make it rigorous. Deﬁne a map H : R → R, where R is a rectangle, as in Figure 9.19. The deﬁnition proceeds in stages: (1) Identify opposite vertical edges to get a cylinder. (2) Squash the top rim to a point to get a cone.

202

Homotopy Versions of Cauchy’s Theorem

Figure 9.19 Topological proof of Proposition 9.17.

(3) Open the cone out ﬂat to get a disc. (4) Distort the disc to get a square. Suppose that γ is a boundary, say γ = ∂φ where φ : R → D. Then φ ◦ H : R → D is a homotopy. The lower edge of R, marked by the heavy line in the ﬁrst diagram, maps to (the image under φ of ) ∂φ . The top edge maps to a point. So γ is homotopic to zero. Conversely, suppose that γ is a closed path homotopic to zero. Then we can deﬁne a map of the cone into D such that the base circle goes to (the image of ) γ . Therefore (reversing the last two steps in the deﬁnition of H) we can map R into D so that its perimeter goes to γ . Hence, up to reparametrisation, γ = ∂φ .

9.8

The Cauchy Theorems Compared A common cause of distress for students of complex analysis is the sudden appearance of a plague of Cauchy Theorems, having several variant hypotheses and a similar variety of conclusions, but all begin derivable from each other. At times like this it may be advisable to seek consolation elsewhere than mathematics: perhaps among the poets. Rudyard Kipling’s In the Neolithic Age makes the point admirably: There are nine and sixty way of constructing tribal lays And – every – single – one – of – them – is – right!

It is much the same with the Cauchy Theorems: all of the different versions are essentially the same. At the heart of all Cauchy-type theorems is the local existence of an antiderivative, Section 8.5. The theorems themselves supply, as hypotheses, conditions that permit this local result to be globalised in some way.

9.8 The Cauchy Theorems Compared

203

For example, in Chapter 8 the local result led to the central Cauchy Theorem 8.8, that if f is differentiable in D and a closed contour does not wind round any point outside D, ± then γ f = 0. This ‘non-winding’ condition ensures that the local pieces of antiderivative ﬁt together well on the global level. The generalised version, with several contours γ1, . . . , γn, is a simple corollary obtained by making cuts between the contours. In this chapter we studied what happens when a contour is deformed continuously. Again the local existence of an antiderivative is involved: it lets us deﬁne the integral of f along an arbitrary path, and it gives the main result of that chapter, that the integral of f along a boundary is zero. This is also a globalisation: associated with any boundary is a map of an entire rectangle, and the local antiderivatives all ﬁt together properly across the image of this rectangle. Moreover, we saw that a path is a boundary if and only if it is homotopic to zero. Since±homotopy leaves the integral round a closed path invariant, this makes the reason why γ f = 0 transparent. We therefore have two main variants of the Cauchy Theorem: one for paths that do not wind round points outside D, and another for paths that are homotopic to zero. But (more Kipling), like the Colonel’s Lady and Judy O’Grady, they are ‘sisters under their skins’. Obviously if γ is homotopic to zero and z0 ± ∈ D, then w(γ , z0)

= 2π1 i

²

1

γ z − z0

dz = 0

because 1/(z − z0 ) is differentiable in D. So the ‘non-winding’ version of Cauchy’s Theorem easily implies the ‘homotopy’ version. It is in fact strictly stronger, in the following sense. The path in Figure 9.20 does not wind round any z0 ± ∈ D, but it is manifestly not homotopic to zero. (However, this is harder to prove than it might appear.) So the ‘non-winding’ hypothesis is weaker, hence applies to more cases. Even this surface difference vanishes when we look a little more deeply, however. Every path that does not wind round any z0 ± ∈ D can be transformed, using a series of cuts whose contributions to the integral cancel, into a closed path (or set of paths) homotopic to zero. For example, Figure 9.20 (left) is so transformed in Figure 9.20 (right). This fact is also non-trivial to prove, but it shows that the practical consequences

Figure 9.20 Left: A path that satisﬁes the non-winding condition but is not homotopic to zero.

Right: Creating cuts in the path makes it homotopic to zero.

204

Homotopy Versions of Cauchy’s Theorem

of the extra generality are largely spurious. (The theoretical consequences are more important: the ‘non-winding’ condition is part of ‘homology theory’, and what we have here is a relation between homotopy and homology. We postpone a homology version of Cauchy’s Theorem to Chapter 16, so that we can move on to more practical issues in Chapters 10–14.)

9.9

Exercises 1. Let D = {z ∈ C : z ± = 0} and Sr (t) = reit (t ∈ [0, 2π ]) for r continuous map φ : R → D, where R is a rectangle, such that

²

∂φ

²

=

f

S1

²

f

S2

= 1, 2. Deﬁne a

f

for any f differentiable in D. Use Theorem 9.6 to deduce that

²

S1

f

=

²

S2

f

Describe a homotopy via closed paths in D from S1 to S2.

2. Let γ1 , γ2 be closed paths in a domain D that are homotopic via closed paths in D. By making a suitable cut σ from γ1 to γ2 , describe a ﬁxed end point homotopy in D from γ1 to σ + γ2 − σ . Draw a picture to illustrate the continuous deformation. 3. Let the boundary ∂φ of a continuous map φ : R → D be subdivided into two subpaths ∂φ = γ1 + γ2. Describe a ﬁxed end point homotopy in D from γ1 to −γ2 . Draw a picture to illustrate the continuous deformation. 4. Draw the semicircle γ (t) = eit (t ∈ [−π/2, π/2]) in D explicit polygonal paths λ1, λ2 from −i to i in D such that

²

λ

f

=

²

γ

= C \ {0}. Deﬁne two

f

is true for all f differentiable in D when λ = λ 1, but false when λ =

5. Let γ : [0, 1]

→ C be given by γ (0) = γ (1) = 0 and ´ t + it sin(π/t) γ (t) = (1 − t) + i(1 − t) sin(π/(1 − t))

for t for t

∈ [0, 21 ] ∈ [ 21 , 1]

Show that γ is a closed path but not a contour, and draw a sketch.

6. Integrate the following functions around the path γ in Exercise 5: (i) cos3 (z2 ) (ii)

∞ ³ n=1

zn /n

(iii) 1/(z −

1 3

2)

λ2.

9.9 Exercises

7. Let f be differentiable in D. For a closed path the integral value Iγ is the complex number Iγ

=

205

γ in D, beginning and ending at z0 ,

²

γ

f

Show that the set of integral values forms a commutative group I( f , D) under the operation Iγ

+ Iδ = Iγ +δ

Determine the group of integral values in the following cases: (i) f (z) = 1/ z, D = C \ {0} (ii) f (z) = cos z, D = C \ {0} (iii) f (z) = 2/ (z − 1) + 3/(z + 1), D = C \ {²1} (iv) f (z) = 1/ (z − 1) + 2/(z + 1), D = C \ {²1}

8. The Fundamental Group. Let D be a domain and z0 ∈ D. For closed paths γ , δ in D that begin and end at z0 , deﬁne γ ³ δ to mean that γ , δ are ﬁxed end point homotopic in D. Show that ³ is an equivalence relation. Let [γ ] denote the equivalence class containing γ , and let π (D, z0) be the set of equivalence classes. Deﬁne the operation ∗ on π (D, z0) by [γ ] ∗ [δ ] = [γ

+ δ]

Check that ∗ is well-deﬁned, and show that π (D, z0) is a group under ∗, specifying the identity element and the inverse of [γ ]. For any other point z1 ∈ D and any path σ in D from z0 to z1 , deﬁne g : π (D, z0) → π (D, z1) by g([γ ]) = [−σ

+ γ + σ]

Show that g is an isomorphism of groups and deduce that π (D, z0 ) is independent of the choice of z0 ∈ D. (This result relies on D being path-connected.) For this reason, the group π (D, z0 ) is usually denoted by π (D) and called the fundamental group of D. 9. Describe (without formal proof, since we have not developed suitable techniques) the fundamental groups of the following domains: (i) C (ii) {z ∈ C : |z| < 1} (iii) {z ∈ C : |1 < z| < 2} (iv) C \ {0} (v) C \ {²1} (vi) C \ Z

10. Let f be differentiable in the domain D, and let γ , δ be closed contours in D beginning and ending at the same point z0 . Show that if γ ³ δ in the sense of Exercise 8, then the integral values Iγ and Iδ are equal. Prove that the map h : π (D) → I( f , D) for which h([γ ]) = Iγ is a well-deﬁned group homomorphism. Describe the homomorphism h for each f in Exercise 7.

206

Homotopy Versions of Cauchy’s Theorem

11. Integrals Along Arbitrary Paths. For ﬁxed z1, z2 ∈ D, suppose that γ0 is a ﬁxed path in D, and γ is a variable path in D, both from z1 to z2 . Show that

²

γ

f

=

²

γ0

f

+ Iσ

for some σ ∈ I( f , D). Prove that if γ is deformed continuously in a homotopy via closed paths in D, then Iσ remains constant. ± For z1 = −i, z2 = i, determine all possible values of γ f for each f , D in Exercise 7.

10

Taylor Series

We now reach a stage in the theory were we can take a great leap forward and show, as promised repeatedly, that any differentiable complex function has a local power series representation. On the slim assumption that the derivative of f exists throughout a domain D, we ﬁnd that near any point z0 ∈ D there is a power series expansion f (z0 + h)

=

∞ ± n=0

an hn

for z0 + h ∈ Nr (z0 )

which is valid – that is, converges – on any disc Nr (z0 ) ⊆ D, see Figure 10.1.

Figure 10.1 The power series round z0 converges on any disc in D.

This theorem releases a tidal wave of theorems, because we can apply the results of Chapter 4 on power series to any differentiable function f . For instance, Corollary 4.21 states that a power series can be differentiated term by terms as many times as we like. Taylor’s Theorem, Corollary 4.22, gives an explicit formula: f (z0 + h) =

∞ f (n)(z ) ± 0 n=0

n!

hn

for z0 + h ∈ Nr (z0 ) ⊆ D

Thus if we insist only that the ﬁrst derivative f ± exists in D, it follows that all higher derivatives exist, and the function is equal to its Taylor series in Nr (z0 ). We derive subtler consequences in this and subsequent chapters, and Cauchy’s Theorem plays a prominent role. It is this sequence of results that gives complex analysis its own special ﬂavour.

208

Taylor Series

Figure 10.2 Contours in the proof of the Cauchy Integral Formula.

10.1

Cauchy Integral Formula The proof that any differentiable function can be expressed as a power series depends on a result of Cauchy, itself of intrinsic interest: 10.1 (Cauchy Integral Formula for a Circle). Let f be differentiable in the disc NR (z0 ) = {z ∈ C : | z − z0| < R}. For 0 < r < R, let Cr be the path Cr (t) = z0 + reit (t ∈ [0, 2π ]). Then for |w − z0 | < r we have

LEMMA

f (w) = Proof. Fix w such that |w − z0 | differentiable in the domain D Let 0 < ε

0, M ≥ 0, we have 0 < | z − w| < δ implies |F(z)| ≤ M The Estimation Lemma, Lemma 6.41, implies that for ε < δ ,

Now limz →w F(z)

³² ³ ³ ³ ³ F(z)dz³ ≤ M · 2πε ³ S ³ ε

(10.1)

209

10.2 Taylor Series

From (10.1),

³ ³² ³ ³ ³ F(z)dz³ ≤ 2M πε ³ ³ C

Since ε is arbitrary, this implies that

r

² Cr

Therefore

² Cr

F(z)dz = 0

f (z) dz = z− w

f (w) dz z Cr − w ² = f (w) z −1 w dz Cr = f (w) · 2π i

In other words,

1

f (w) =

10.2

²

²

2π i

Cr

f (z) dz z −w

Taylor Series Using the Cauchy Integral Formula we can now expand f (z0 + h) as a power series in h, with coefﬁcients expressed as integrals. LEMMA

10.2. Let f be differentiable in NR (z0).Then there exist a n f (z0 + h) =

where the series converges absolutely for | h|

∞ ± n=0

∈ C such that

an hn

< R. Further, if C r (t) = z0 + reit (t ∈ [0, 2π ])

then an

=

1 2π i

² Cr

f (z) dz (z − z0)n+1

Proof. Fix h with 0 < |h| < R and initially suppose that r satisﬁes | h| Figure 10.3. The Cauchy Integral Formula gives f (z0 + h) =

1 2π i

=

1 2π i

=

m ± n=0

²

Cr

²

Cr

f (z) dz z − (z0 + h)

´

f (z)

m

k for all z on CR . Then

³ ³ ³ 1 ³ ³ ³ ³ P(z) ³

and Cauchy’s Estimate gives

≤ k2n

for all z on CR

³ ³ ³ 1 ³ 2 ³ ³ ³ P(z) ³ ≤ kn

By Liouville’s Theorem, 1 /P(z) is constant, so P(z) is constant. But this contradicts n ≥ 1. Therefore P(w) = 0 for some w ∈ C. It follows in the usual way that any polynomial P(z) of degree n, with complex coefﬁcients, can be expressed as a produce of terms of degree 1: where the αj

10.5

∈ C.

P(z) = (z − α1)(z − α2) · · · (z − αn )

Zeros We now broaden our perspective from polynomials, and look at the zeros of arbitrary differentiable functions. 10.10. A zero of a differentiable function f : D for which f (z0 ) = 0.

D E FI N I T I O N

→ C is a point z0 ∈ D

Expanding f in a Taylor series about the zero z0 , we have f (z) =

∞ ±

n= 0

an(z − z0 )n

for |z − z0|

0 such that D E FI N I T I O N

0 < |z − z0 |

LEMMA

< δ implies f (z) ²= 0

10.13. A zero of ﬁnite order is isolated.

Proof. Write f (z0 ) = (z − z0 )m g(z) for |z − z0| < R, where g is differentiable and g(z0) ² = 0. Then g is continuous at z0 , so, taking ε = 21 | g(z0 )| there exists δ > 0 such that

|z − z0 | < δ implies |g(z) − g(z0)| < ε Therefore, when |z − z0 | < δ we have |g(z)| ≥ ||g(z0 )| − |g(z0) − g(z)|| > 2ε − ε = ε In particular, g(z) ² = 0. But if 0 < |z − z0 | < δ then |z − z0 |m ² = 0, so f (z) = (z − z0) m g(z) ² = 0. 10.14. Let S be a set of zeros of a differentiable function f in D, having a limit point z0 ∈ D. Then f is identically zero in any disc NR (z0 ) ⊆ D. C O RO L L A RY

216

Taylor Series

Proof. Because z0 is a limit point of S, there is a sequence {zn }n≥1 in S that tends to z0 . Then f (z0 ) = limn→∞ f (zn) = 0. So z0 is a zero of f that is not isolated. Hence it does not have ﬁnite order, so f (z0 + h) =

∞ ± n=0

an hn

in any disc NR (z0) ⊆ D, where all an are zero. From this we deduce: 10.15. If f is differentiable in a domain D and S is a set of zeros of f with a limit point z0 ∈ D, then f is identically zero on D.

P RO P O S I T I O N

Proof. Corollary 10.14 gives f (z) = 0 for z in any disc NR (z0 ) ⊆ D. For any other z ∈ D, choose a path γ : [a, b] → D from z0 to z, Figure 10.4. We show that f (γ (t)) = 0 for all t ∈ [a, b]. By continuity, we can ﬁnd δ > 0 such that a ≤ t < a + δ implies γ (t) ∈ NR (z0 ) so f (γ (t)) = 0 for a ≤ t < a + δ . Let s be the least upper bound of those x ∈ [a, b] such that f (γ (x)) = 0 for a ≤ t < x. Then a + δ ≤ s ≤ b. By continuity, f (γ (s)) = 0. If s < b then γ (s) is a non-isolated zero, so f is identically zero in a neighbourhood of γ (s). Then there exists an interval [s, s + κ ] with κ > s, on which f (γ (t)) = 0, contradicting the deﬁnition of s. Hence s = b, so f (z) = f (γ (b)) = 0. An important consequence is: 10.16 (Identity Theorem). If f and g are differentiable in a domain D and f (z) = g(z) for all z ∈ S ⊆ D where S has a limit point in D, then f = g throughout D.

THEOREM

Proof. Apply Proposition 10.15 to f

Figure 10.4 Proof of Proposition 10.15.

− g.

10.6 Extension Functions

217

Example 10.17. It is essential for the relevant limit point of S to be in D: if not, the theorem can be false. Let

= sin(1/z) g(z) = 0 = g(z) for z = ³1/(nπ ) and z0 = f (z)

in D = C \ {0}. Then f (z) S = {³1/(nπ )}, but f ² = g in D.

10.6

0 is a limit point of

Extension Functions Combining the Identity Theorem with power series leads to a method that can often extend the domain of deﬁnition of a complex differentiable function. D E FI N I T I O N 10.18. A function f : D → C is an extension function of h : S S ⊆ D and f (z) = h(z) for all z ∈ S. (We also say that f extends h.)

→ C if

= 1/(1 − z) on D = C \ {1} is an extension function of h(z) = = { ∈ C : |z| < 1}.

Example 10.19. f (z) ∑∞ n z n=0 z on S

Suppose that D is a domain and S ⊆ D has a limit point in D. The Identity Theorem shows that if a function f : S → C has a differentiable extension function g : D ⊆ C, then the extension is unique. As an application, suppose that f : D → C is differentiable and consider the Taylor ∑∞ n expansion f (z) = n=0 a n(z − z0 ) of f in a disc Nr (z0) ⊆ D. Then f is the only possible extension of the Taylor expansion to the whole of D. This means that although the Taylor expansion may not converge on the whole of D, it contains all the information needed to determine f uniquely throughout D. We use this fact to great advantage in Chapter 14. Example 10.20. Consider the power series

z2 2

n

+ · · · + ( −1)n zn + · · · (10.4) on the small disc S = {z ∈ C : | z| < 1/1 000 000}. We can certainly extend this function z−

to a larger disc, because the radius of convergence of the power series is 1. However, we can extend beyond this. For instance, if K = {t ∈ R : t ≤ −1} and D = C \ K, then f (z) = log(1 + z) is the (unique) differentiable extension of the power series to D, Figure 10.5.

218

Taylor Series

Figure 10.5 Extending the power series (10.4) to

C \ K.

The set S need not be a domain. In particular, it may be a subset of R:

=

Example 10.21. f (z) sin z for z C is the unique differentiable extension function of f (x) sin x for x R. This is a consequence of the usual convergent Taylor expansion in the real case.

=

Of course, if D is a domain then it is open, so D ∩ R is open in R. A differentiable function f : D → C has a power series expansion in a neighbourhood of any x0 ∈ D ∩ R. So if a real function h : S → R extends to a differentiable complex function on a domain, it must already have a power series expansion about any point in S ⊆ D ∩ R. Thus the only real functions that extend to differentiable complex functions are real analytic functions. We may take S to be even more restricted, provided that it has at least one limit point in D:

/ = /

Example 10.22. If f (1 n) 1 n2 for all positive integers n, then f (z) analytic extension of f to the whole complex plane, because

= z2 is the unique

S = {1/n : n is a positive integer} has the limit point 0 ∈ C. The Taylor expansion is valid on discs (Theorem 10.3), and may therefore be used to deﬁne extension functions. It is therefore easy to see that the radius of convergence of the Taylor expansion of a function f about a point z0 is equal to the distance from z0 to the nearest point z1 at which no differentiable extension function of f may be deﬁned. Such a point is called a singularity of f , see Chapter 11. Singularities are obstacles that determine the radius of convergence of a Taylor series. (There is no relation between this use of ‘singularity’ and the term ‘singular’ in Deﬁnition 6.20. ‘Singular’ is just one of those words that get overused in mathematics.)

10.7 Local Maxima and Minima

10.7

219

Local Maxima and Minima The complex numbers are not ordered, so we cannot speak of maxima and minima of a complex function f . We can, however, consider maximum and minimum values of the modulus | f |, since this is real. D E FI N I T I O N 10.23.

If f : D

→ C, then

(i) The modulus | f | has a local maximum at z0 ∈ D if there exists ε > 0 such that N ε (z0) ⊆ D and | f (z)| ≤ | f (z0 )| for all z ∈ Nε (z0 ). The local maximum is strict if | f (z)| < | f (z0)| for all z ∈ Nε (z0) \ {z0}. (ii) The modulus | f | has a local minimum at z0 ∈ D if there exists ε > 0 such that N ε (z0) ⊆ D and | f (z)| ≥ | f (z0 )| for all z ∈ Nε (z0 ). The local minimum is strict if | f (z)| > | f (z0)| for all z ∈ Nε (z0) \ {z0}. The problem of ﬁnding local maxima of | f | in a domain turns out to be easy (or impossible, depending on how you look at it, since there are none.) P RO P O S I T I O N 10.24. A differentiable function has no strict local maximum of its modulus in its domain. If it has a local maximum in its domain, it is constant.

Proof. Suppose that f is differentiable in D and let z0 be a local maximum in D. Then | f (z)| ≤ | f (z0 )| for all z ∈ Nε (z0). For 0 < r < ε , the circle Cr (t) = z0 +reit (t ∈ [0, 2π ]) lies inside Nε (z0 ), so | f (z0 + reit )| ≤ | f (z0 )| for all t ∈ [0, 2π ]. The Cauchy Integral formula gives f (z0 ) =

= = so that

| f (z0 )| ≤

1 2π

² 2π 0

1 2π i

²

Cr

²

f (z) dz z − z0

2π f (z + reit ) 1 0 ireit dt 2π i 0 reit ² 2π 1 f (z0 + reit )dt 2π i 0

| f (z0 + re )|dt ≤ it

Hence

| f (z0) | = 21π

² 2π 0

1 2π

² 2π 0

| f (z0 )|dt ≤ | f (z0)|

| f (z0 + reit )|dt

(10.5)

If the strict inequality | f (z0 + reit )| < | f (z0)| were to hold for any t ∈ [0, 2π ], then by continuity it would also hold in a small interval, giving the strict inequality 1 2π contradicting (10.5).

² 2π 0

| f (z0 + reit )|dt < | f (z0 )|

220

Taylor Series

Hence | f (z0 + reit )| = | f (z0) | for all t ∈ [0, 2π ]. This holds for any r < ε , so | f | is constant in Nε (z0 ). By Proposition 4.15, f is constant in Nε (z0 ). The Identity Theorem now shows that f is constant throughout D. For the complementary notion of a local minimum of | f |, it is clear that if f is nonconstant and has a zero at z0 then z0 is isolated and | f | has a strict local minimum there. If f is non-zero on D, observe that | f | has a strict local minimum if and only if |1/f | = 1/| f | has a strict local maximum. We can apply Proposition 10.24 to 1/f to get: P RO P O S I T I O N 10.25. A differentiable function that does not equal zero anywhere in its domain has no strict local minimum of its modulus in its domain. If it has a local minimum in its domain, it is constant.

10.8

The Maximum Modulus Theorem The question of maxima or minima of | f | on an arbitrary subset of the domain of f is somewhat different. 10.26. Let f : D → C be differentiable in a domain D, and let S Then | f | has a local maximum on S at z0 ∈ S if

D E FI N I T I O N

⊆ D.

(i) z0 is a limit point of S. (ii) For some ε > 0, | f (z)| < | f (z0 )| whenever z ∈ Nε (z0) ∩ S. Condition (i) is essential, for otherwise some neighbourhood Nε (z0 ) contains no points of S other than z0 , and then condition (ii) is vacuously true.

=

={ ∈ =

| | ≤ 1}, then | f (x + iy)| = ex and | f | has

Example 10.27. If f (z) ez and S z C: z a local maximum on S at the point z0 1.

Using Proposition 10.24, we see that if Nε (z0) ⊆ S, then | f | cannot have a strict local maximum at z0 . To explore this further, we need: 10.28. A point z0 is a boundary point of S if every neighbourhood of z0 contains a point in S and a point not in S, other than z0 itself. In other words, a boundary point is a limit point both of S and its complement C \ S. The boundary ∂ S of S is its set of boundary points.

D E FI N I T I O N

={ ∈ { ∈ | |= }

| | ≤ 1} and of of its complement

Example 10.29. The boundary of S z C: z z C: z 1 is the circle z C : z 1.

{ ∈

| |> }

We can now rephrase Proposition 10.24 to give:

10.9 Exercises

221

T H E O R E M 10.30 (Maximum Modulus Theorem). If a differentiable function is not constant, then any local maximum value of its modulus on an arbitrary subset of its domain occurs on the boundary of that set.

We also have from Proposition 10.25: T H E O R E M 10.31 (Minimum Modulus Theorem). If a differentiable function is not constant, then any local minimum value of its modulus on an arbitrary subset of its domain occurs either at a zero of the function, or on the boundary of that set.

=

={ ∈

||≤ }

Example 10.32. If f (z) z2 on the set S z C: z 1 , then the maximum values 2 2 of f (x iy) x y occur all round the boundary of S, while the minimum value occurs at the origin.

|

10.9

+ |= +

Exercises 1. Find the Taylor series at 0 of f (z) = Log(1 + z), where Log is the principal value. What is the disc of convergence? Answer the same questions for g(z) = exp(α Log(1 + z)), where α ∈ C. 2. Find the ﬁrst three terms and radius of convergence for the Taylor series at 0 of F(z) = [1 + Log (1 − z)]−1. 3. Taylor expand the following functions around 0, and ﬁnd the radius of convergence. sin2 z z2 (z + 2)−2 + b)−1 (a, b ∈ C, b ²= 0) º(az z exp(w2 )dw ´0 (sin z)/z (z ² = 0) (v) (z = 0) ºz1 (vi) 0 (sin w)/w dw

(i) (ii) (iii) (iv)

4. Deﬁne the numbers cn by the Taylor series sec z =

∞ ± n=0

(−1)n

c2n 2n z (2n)!

Prove that

=1 ¶ 0 = c0 + c2

c0

2n 2

·

+ c4

2n 4

·

+ · · · + c2n

Show that c2n is always an integer and calculate it for n ≤ 5. 5. Let

(1 − z − z2 )−1

=

±

Fn zn

2n 2n

·

222

Taylor Series

Prove that F0

= F1 = 1

= Fn−1 + Fn−2

Fn

(n ≥ 2)

This is the recursive deﬁnition of the Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, . . .. By expanding (1 − z − z2 )−1 in partial fractions, prove that Fn

1

=√

5

⎡¸ √ ¹n+1 ¸ √ ¹n+1 ⎤ 1+ 5 ⎣ − 1− 5 ⎦ 2

2

6. Investigate analogous results to Exercise 5 on the expansion of (1 for a, b ∈ C. 7. If

¶ exp

·

z

1−z

prove that a0

=1 ∑

an

=

=

±

− az − bz2)−1 ,

an zn

¶ · n ± 1 n−1 s!

s−1

< R. Let ω = e2π i /3 and deﬁne g(z) = 31 ( f (z) + f (ωz) + f (ω 2z))

8. Let f (z) have Taylor series

Show that

anzn for | z|

s=1

g(z) =

±

for | z| < R. Find similar expressions for 1 + ω + ω2 = 0.)

a3n z3n

a3n+1 z3n+1 and

a3n+2 z3n+2 . (Hint:

9. Deﬁne three functions by

α (z) =

± z3n

(3n)!

β(z) =

± z3n+2

(3n + 2)!

γ (z) =

± z3n+1

(3n + 1)!

Prove the series converge for all z. Using Exercise 8, prove the following:

α± (z) = β(z) β± (z) = γ (z)√ γ ± (x) = α(z) α(z) = 31 [ez + 2e−z/2 cos(z 3/2)]; ﬁnd similar expressions for β(z), γ (z). α(z + w) = α(z)α(w) + β (z)γ (w) + γ (z)β (w) α3 (z) + β3 (z) + γ 3(z) = 1 α(z) = β(z) ⇐⇒ z = (3n − 1) · 32√π3 (n ∈ Z) ∑ Generalise Exercise 8 to ﬁnd an expression for apnzpn (p = 2, 3, . . .), and (harder) ∑ for apn+k zpn+k (p = 2, 3, . . . ; k = 0, 1, . . . , p − 1). Show that the Cauchy estimate is an equality if and only if f (z) = Kzn for some K ∈ C, n = 1, 2, 3, . . ..

(i) (ii) (iii) (iv) (v) 10. 11.

12. Let D be a disc centre z0 , and let f be differentiable in a domain containing D. Prove that the ‘mean value’ of f (z) as z runs over ∂ D (deﬁned by a suitable integral) is equal to f (z0).

10.9 Exercises

223

13. Let f be differentiable throughout C, and suppose that | f (z)| ≤ K |z| c for a real constant K and positive integer c. Prove that f is a polynomial function of degree ≤ c.

14. Let f and g be differentiable on the strip D = {z ∈ C : −2 < im z < 2}. Suppose that f (z) = g(z) for all z such that |z| < 0.01. By considering Taylor expansions ﬁrst about 0, then 1, and so on by induction, prove that f (z) = g(z) on D. 15. Suppose that f (z) show that

= ∑ an (z − z0)n in a disc D centre z0, radius R. If 0 ≤ r < R, 1 2π

² 0

2

| f (z0 + reiθ )|2 dθ =

±

|an|2r 2n

(Parseval’s Inequality). Hence show that

±

|an |2 r2n ≤ M(r) = sup | f (z0 + reiθ )| θ

Use this to give an alternative proof of the Maximum Modulus Theorem.

16. If p(z) is a polynomial of degree n, show that for each R > 0 the ‘level curve’ of |p(z)|, deﬁned to be {z ∈ C : |p(z) | = R}, has at most n connected components.

17. Suppose that x2 + y2 ≤ a2 . Prove that (1 + x)2 + y2 attains its maximum value when x = a, y = 0. (Hint: apply the Maximum Modulus Theorem to 1 + z on the disc |z| = a.) 18. Suppose that x2 + y2 ≤ 1. Prove that (x2 − y2 value when x = 0, y = ³1.

− 1)2 + 4x2 y2 attains its maximum

19. If f is differentiable in a domain D, prove that the zeros of f are either all of ﬁnite order and isolated, or f is identically zero on D. 20. Let f (x) =

² ∞ x

t −1 ex −t dt

where x is real and positive. By repeated integration by parts, show that if hn(x) = (−1)n n! x−(n+1) then f (x)

= h0 (x) + h1 (x) + · · · + hn−1 (x) + (−1)

Show that the series

∞ ± n=0

diverges for all x. Show also that

n

² ∞

hn(x)

³ ³ N ³ ³ ± ³ f (x) − ³ hn (x) ³ < N !x−(N+1) ³ ³ ³ n=o

x

ex t−(n+1)dt

(10.6)

224

Taylor Series

so that for large enough x the series (10.6) provides an arbitrarily good approximation to f (x), even though it diverges. (The catch is that, the better the approximation required, the larger x has to be. No particular choice of x gives an arbitrarily good approximation if we take N large enough. A series with this property is called an asymptotic series.)

11

Laurent Series

The Taylor series expansion is too limited for many applications. A useful generalisation was given by Laurent in 1843. Weierstrass may have discovered it two years earlier, but his paper was not published until after his death. Laurent considered ‘power series’ involving negative powers as well as positive. The beneﬁts that accrue are hinted at by 2 the following example. The function f (z) = e−1/z is very badly behaved as regards Taylor series expansion. We have seen that, restricted to the real line, its Taylor series about the origin is 0 + 0x + 0x2 + · · · , which does not converge to f (x). On the complex plane it is, if such a statement makes sense, even less capable of being represented by a Taylor series. The natural series representation is obtained by starting with the Taylor series for ez and replacing z by −1/ z2, giving f (z)

= 1 − z−2 + 21! z−4 + 31! z−6 + · · ·

and this is a series of the ‘negative powers’ type. It converges for all z for which −1/z2 is deﬁned, namely z ± = 0.

11.1

Series Involving Negative Powers The general series of this type can be written in the form

∞ ±

n=−∞

an(z − z0)n

which is to be thought of as a compact notation for

²∞ ± n=0

an(z − z0)

³

n

+

²∞ ± n=1

a−n (z − z0)−n

³

and hence converges if and only if the two series in parentheses converge. We know that power series converge inside a disc. Consequently power series with negative powers 2 alone should converge outside a disc. For instance, that for e −1/ z converges outside the disc | z| = 0. Therefore those with both positive and negative powers should converge in a region between two concentric circles. Recall that such a region is called an annulus. More precisely, if R1, R2 ∈ R with 0 ≤ R1 < R2 ≤ ∞, and z0 ∈ C, then

{z ∈ C : R1 ≤ |z − z0| ≤ R2}

226

Laurent Series

is an annulus. We begin with an existence theorem for series expansions of the above kind. 11.1 (Laurent’s Theorem). If f is differentiable in the annulus R1 z0 | ≤ R2 where 0 ≤ R 1 < R2 ≤ ∞, then

THEOREM

f (z0 + h) =

∞ ±

an hn +

n=0

∞ ±

bn h−n

n=1 − bn h n converges for | h|

where an converges for | h| < R2 , particular both sides converge in the interior of the annulus. Further, if Cr (t) = z0 + reit , (t ∈ [0, 2π ]) then hn

an bn

1

= 2π i 1

= 2π i

´

Cr

´

Cr

>

R1 , and in

f (z) dz (z − z0 )n+1

f (z)(z − z0 )n−1 dz

Before giving the proof: 11.2. In the more compact notation we set cn (n ≥ 1). Then REMARK

f (z0 + h)

≤ |z −

∞ ±

=

n=−∞

= an (n ≥ 0) and c−n = bn

cnh n

which converges in the interior of the annulus, and cn for all n ∈ Z. Proof of Theorem 11.1. If R1

= 2π1 i

´

Cr

f (z) dz (z − z0 )n+1

< |h| < R2 , choose r1 , r2 such that R1 < r1 < |h| < r2 < R2

and let Cr1 (t) = z0 + r1eit Cr2 (t) = z0 + r2e

as in Figure 11.1. We show ﬁrst that f (z0 + h)

= 2π1 i

´ Cr 1

it

∈ [0, 2π ]) (t ∈ [0, 2π ])

(t

1 f (z) dz − (z − (z0 + h)) 2π i

To do so, enclose z0 + h in a small circle Sε (t)

= z0 + εeit

Then the function F(z) =

´ Cr 2

f (z) dz (z − (z0 + h))

(t ∈ [0, 2π ])

f (z) z − (z0 + h)

(11.1)

11.1 Series Involving Negative Powers

227

Figure 11.1 An annulus, showing paths used to prove Theorem 11.1.

is differentiable in S = {z ∈ C : R1

< |z − z0| < R2 , z ±= z0 + h}

and the contours −C r2 , Cr1 , Sε satisfy the hypotheses of Theorem 8.9. Therefore

´

−Cr2 Hence

´ Sε

F(z)dz +

F(z)dz =

´

Cr 1

´ Cr 2

F(z)dz +

´

F(z)dz −

F(z)dz = 0

´ Cr 1

F(z)dz

But by Cauchy’s Integral Formula,

´

F(z)dz = 2π i · f (z0 + h)

giving (11.1). (Alternatively we can make cuts and integrate round the two halves, as shown in Figure 11.2. The parts of the contours along the cuts cancel in pairs when integrated.) All we need do now is work out the two integrals in (11.1) as power series, and calculate the coefﬁcients. Unfortunately this is the longer part of the proof, although the calculations are routine. First, we choose ρ1, ρ2 with r1 < ρ 1

< |h| < ρ2 < r2

which enforces the conditions: (i) | z − (z0 + h)| (ii) | z − (z0 + h)|

> r2 − ρ2 for z on Cr ; and > ρ1 − r1 for z on Cr . 2

1

228

Laurent Series

Figure 11.2 Alternative proof using cuts.

As in the proof of the Taylor series expansion, Lemma 10.2, we get

´

1

2π i

± n f (z) dz = an h z − (z0 + h) n=0

Cr1

for |h| < ρ 2, where

´

1

an =

2π i

f (z) dz (z − z0 )n+1

Cr 1

The treatment of the second integral is similar, but will be given in full. Since 1 h

n−1

+ z −h2z0 + · · · + (z − hz0n )

(summing a geometric series) we have 1

− 2π i

´

Cr2

1 f (z) dz = (z − z0 − h) 2π i

=

n ± m =1

1 f (z) h Cr

Bn

+ ··· +

2

0

(z − z0)n−1 hn

n − hn (z(z−−zz0 )− h) dz 0

bm h−m − B n

where bm

0

µ

´

n − hn (z(z−−zz0 )− h) = z − −z 1− h

= 21π i = 21π i

´ Cr2

´

Cr2

f (z)(z − z0)m −1dz f (z)(z − z0 )n hn(z − z0 − h)

Finally we estimate the size of Bn . There exists M > 0 such that | f (z)| ≤ M on Cr1 since this circle is closed and bounded. By (i) above, | z − z0 − h| > ρ1 − r1 . Also |h| > ρ1 , |z − z0| = r1. Hence n

·

|Bn| ≤ 21π ρn (ρMr−1 r ) · 2π r1 = ρ Mr−1r ρr1 1 1 1 1 1 1

¸n

11.1 Series Involving Negative Powers

229

which tends to 0 as b = n → ∞ , since r1/ρ 1 < 1. It follows that 1

− 2π i

∞ ± f (z) dz = bm h−m (z − z0 − h)

´

m=1

Cr 2

To ﬁnish the proof we must replace C r1 and Cr2 by Cr in the expressions for an and bn. Since all three paths are homotopic inside the annulus, or by using cuts, this is immediate. Note that we can no longer assert that an = f (n) (z0)/n! since f (z) need not be differentiable for | z − z0 | < R1 under the stated hypotheses. D E FI N I T I O N

series

11.3. The Laurent series or Laurent expansion of f (z) about z0 is the

∞ ± n=−∞

cn hn

where h = z − z0 and cn is deﬁned above. THEOREM

11.4. The Laurent expansion of f about z 0 is unique.

Proof. Suppose that f (z) = Then (z − z0) −m−1f (z) =

∞ ± n=−∞

∞ ± n=−∞

dn (z − z0 )n

dn (z − z0 )n−m−1

= f1 (z) + z −dmz + f2(z) 0

where f1 (z) = f2 (z) =

m −1 ± n=−∞

∞ ±

n=m+1

dn (z − z0 )n−m−1 dn (z − z0 )n−m−1

But each of f1 , f2 has an antiderivative in R1 term by term. That is, set F 1(z)

=

F 2(z)

=

< |z − z0| < R2 , because we can integrate

m −1 ±

1

dn (z − z0)n−m

∞ ±

1

dn(z − z0)n−m

m−n n=−∞ n=m+1

m− n

It is easy to check that the series converge absolutely for R1 F1² (z) = f 1(z), F²2(z) = f2 (z). Hence

< |z − z0 | < R2 and that

230

Laurent Series

´ Cr

so dm

f (z)(z − z0)−m −1dz =

´ Cr

= cm as deﬁned above.

Example 11.5. Let f (z)

= ez + e1/z. We know that ez

=

e1/z

=

so

∞ 1 ± n=0

n!

zn

n=0

n!

∞ ± m=∞

where

= 1/m! =2 = 1/(−m)! and the expansion is valid for z ± = 0. cm c0 cm

cm zm (m ≥ 1) (m ≤ −1)

= 1z + 1 −1 z . In a similar way, f (z) = ∑ cmzm where =0 c m =1 cm

Example 11.7. f (z)

for all z

∞ 1 ± z−n for all z ± = 0 f (z) =

Example 11.6. f (z)

dm dz = 2π idm z − z0

(m < −1)

(m ≥ − 1)

= z −1 1 − z −1 2 . Writing this as 1 + z(1 − 1/z) 2(1 − z/2) 1

= ∑ cmzm where cm = 1 (m < −1) cm = 2−(m +1) (m ≥ 0) valid in the annulus 1 < |z| < 2.

we obtain an expansion f (z)

11.2

Isolated Singularities If f is differentiable in a punctured disc 0 < | z − z0 | < R

where R

>0

11.2 Isolated Singularities

231

we say that z0 is an isolated singularity of f . We can use the Laurent expansion to study such singularities. There is a Laurent series f (z) =

∞ ± n=0

an (z − z0 )

n

+

∞ ± n=1

bn(z − z0) −n

valid for 0 < |z − z0 | < R. This series can behave in three radically different ways: (1) All bn = 0. If we deﬁne f (z0) = a0 we obtain a function that is differentiable on the whole disc |z − z0| < R, with Taylor series

∞ ± n=0

an(z − z0)n

In this case z0 is said to be a removable singularity. It arises from our choice of a domain of deﬁnition of f , rather than from any intrinsic feature of f . For example, consider f (z) = Around z0

= 0 we have

sin z z

f (z) = 1 −

z2 3!

(z ± = 0)

4

+ z5! − · · ·

so by deﬁning f (0) = 1 we get a function differentiable for all z ∈ D. (2) Only ﬁnitely many bn ± = 0. Then f (z) =

bm (z − z0)m

+ · · · + z −b1z + 0

∞ ± n=0

an (z − z0 )n

where bm ± = 0. In this case we say that f has a pole of order m at z0. For example f (z) = z−4 sin z (z ± = 0)

= z13 − 31!z +

∞ ± n=0

(−1)n

z2n+1 (2n + 5)!

has a pole of order 3 at z0 = 0. (2) Inﬁnitely many bn ± = 0. Then we say that z0 is an isolated essential singularity. For example f (z) = sin(1/ z) (z ± = 0) = 1z − 3!1z3 + 5!1z5 − · · ·

has an isolated essential singularity at z0 = 0. We investigate the behaviour of a differentiable function near an isolated singularity, according to these three possibilities.

232

Laurent Series

11.3

Behaviour Near an Isolated Singularity Removable singularities are trivial and uninteresting – which make it all the more important to recognise them. The next lemma is usually sufﬁcient for this purpose. 11.8. The following are equivalent for a function f that is differentiable for 0 < |z − z0| < R:

LEMMA

(i) z0 is a removable singularity of f . (ii) lim z→z 0 f (z) exists and is ﬁnite. (iii) There exist M > 0, δ > 0 such that | f (z) | < M for 0 < |z − z0|

< δ. Trivially (i) ⇒ (ii) ⇒ (iii), so we need prove only that (iii) ⇒ (i).

Proof. Suppose (iii) holds, and take a Laurent expansion f (z) =

∞ ± n=0

an (z − z0 )n +

Now bn where Cr (t)

1

= 2π i

´ Cr

∞ ± n=1

bn(z − z0)−n

f (z)(z − z0 )n−1 dz

= z0 + reit (t ∈ [0, 2π ]) and 0 < r < R. So |bn| ≤ 21π Mrn−1 · 2π r = Mrn

if we let r → 0 it follows that |bn| = 0 for all n ≥ 1. Thus z0 is a removable singularity and (i) holds. We immediately deduce the useful: 11.9. If any coefﬁcient bn open disc with centre z0 .

C O RO L L A RY

±= 0 (n ≥ 1) then f

is unbounded on every

For example, if f (z) = then lim f (z) = lim

z →0

z→0

· ¹

z2 (ez − 1) sin z

¸·

z

ez − 1 z 2!

z sin z

= zlim 1+ + ··· →0 =1

(z ± = 0)

¸

º−1 ·

1−

z2 3!

+ ···

¸−1

so z0 = 0 is a removable singularity. There is a similar, slightly more complicated, criterion for poles.

11.3 Behaviour Near an Isolated Singularity

P RO P O S I T I O N 11.10. If f is differentiable for 0 order m at z 0 if and only if

lim (z − z0 )m f (z)

z→0

< |z − z0| < R, then f has a pole of

= l ±= 0

Proof. If f has a pole of order m then (z − z0 )m f (z) so that

= bm + · · · + b1 (z − z0 )m−1 + lim (z − z0)m f (z) = bm

z →0

Conversely, if the limit is l ± = 0, then g(z) at z0 by Lemma 11.8, so there is a series g(z) = valid for |z − z0 |

where a0

a0 (z − z0) m

∞ ±

f (z) =

n=0

+ · · · + za−m−z1 + 0

5z + 3

n=0

an (z − z0 )n

±= 0

an(z − z0)n

±= 0, so f has a pole of order m at z0.

For example, consider

∞ ±

= (z − z0 )mf (z) has a removable singularity

< R, and a0 = l ±= 0. But now f (z) =

233

(1 − z)3 sin2 z

∞ ± n=0

an+m (z − z0 )n

(0 < |z| < 1)

Then limz →0 z2f (z) = 3 ± = 0, so there is a double pole (pole of order 2) at the origin. Also, limz →1(z − 1)3 f (z) = −8/ sin2 (1) ± = 0, so there is a triple pole (pole of order 3) at z0 = 1.

The function f (z) has a pole of order m at z0 if and only if 1/f (z) has a removable singularity at z0 which, if removed, gives rise to a zero of order m at z0 . This in particular occurs when 1/f (z) has a zero of order m at z0 . C O RO L L A RY 11.11.

Proof. If f (z) has a pole of order m at z0 then g(z) = (z − z0 )m f (z) has a removable singularity at z0. Further, these exists δ > 0 such that g(z) ± = 0 for |z − z0| < δ. So 1/f (z) = (z − z0)m / g(z)

and 1 /g(z) is differentiable for | z − z0| < δ . Therefore, with the singularity removed, 1/f (z) has a zero of order m at z0. The converse is proved by an almost identical argument. C O RO L L A RY 11.12.

If f (z) has a pole of order m at z0 , then lim | f (z)| = +∞

z →z0

234

Laurent Series

Proof. lim | f (z)| = lim |g(z)/(z − z0 )m |

z →z 0

z →z 0

−m = |l| · zlim → z | z − z0 | =+∞ 0

Thus near a pole the behaviour of f is really quite good. However, near an isolated essential singularity the behaviour is much wilder. The following result is classical: 11.13 (Weierstrass–Casorati Theorem). In every neighbourhood of an isolated essential singularity z0 a differentiable function f takes values arbitrarily close to any assigned complex number. Speciﬁcally, given r > 0, ε > 0, and w ∈ C there exists z1 such that | z1 − z0| < R and | f (z1 ) − w| < ε.

THEOREM

Proof. We have f (z) =

∞ ± n=0

an (z − z0 )

n

+

∞ ± n=1

bn(z − z0)−n

with inﬁnitely many bn ± = 0. If we deﬁne φ (z) = f (z) − w then the Laurent series of φ (z) differs from that of f (z) only in the coefﬁcient a0 , which becomes a0 − w. Hence φ (z) also has an isolated essential singularity at z0, so we need prove only that φ(z) can be made arbitrarily small in any neighbourhood of z0 . That is, there exists z1 with 0 < |z1 − z0 | < r

|φ(z1 )| < ε

If z0 is a limit of zeros of φ this is trivial. Failing this, there exists ρ > 0 such that φ (z) ±= 0 for 0 < |z1 − z0| < ρ . Either there exists z1 with 0 < |z1 − z0| < r and 1/φ(z1 ) > 1/ε, or else 1/φ (z) is bounded for 0 < |z1 − z0| < r. If the former, then the theorem is proved. If the latter, then 1 /φ(z) has a removable singularity at z0 by Lemma 11.8, so by Corollary 11.11 φ (z) has at worst a pole at z0 , which contradicts z0 being essential. Thus the latter case cannot occur. In point of fact, a stronger and more satisfying result is true. Émile Picard (1856– 1941) proved that in every neighbourhood of an isolated essential singularity, a differentiable function f takes every value, with at most one exception. For instance, sin(1 /z) takes every value in 0 < | z| < r for any r > 0, as can easily be veriﬁed. The exception can occur: e1/z misses out the value 0 but attains all others in 0 < |z| < r for any r > 0. But the proof of Picard’s Theorem requires machinery (elliptic modular functions) considerably beyond the reach of this text.

11.4

The Extended Complex Plane, or Riemann Sphere In real analysis it is standard to extend the real line R by adjoining two points at inﬁnity, +∞ and −∞ . We then have a simple way to discuss and visualise limits such as limx →+∞ 1/x and limx→−∞ ex .

11.4 The Extended Complex Plane, or Riemann Sphere

235

Sometimes it is more helpful to consider these two points to be identical. For example, the graph of y = 1/x is a hyperbola, which heads off to ³∞ along both axes. If we identify +∞ with −∞ (known as the one-point-compactiﬁcation of R) the extended hyperbola closes up to form a closed curve, like an ellipse. Both of these curves are conic sections, and in projective geometry they are projectively equivalent. So the projective line extends R by adjoining a single point at inﬁnity. There is a similar extension of the complex plane, which lets us describe the behaviour of a complex function ‘at inﬁnity’ by adjoining to C a single extra point ‘∞ ’. We adjoin just one point in the complex case because the distinction between the two ends of the real line becomes blurred when we deal with the whole plane. For instance, if we rotate C continuously through π , +∞ and −∞ in R swap places. The method for adjoining this extra point was introduced by Bernhard Riemann, and has an elegant geometric realisation. Think of C as being embedded in the (x, y)-plane in R3 , so that a point x + iy ∈ C is identiﬁed with (x, y, 0) ∈ R3 . Let S2

= {(ξ , η, ζ ) ∈ R3 : ξ 2 + η2 + ζ 2 = 1}

be the unit sphere. A line joining the North pole (0, 0, 1) to (x, y, 0) cuts S2 in a unique point (ξ , η , ζ ), giving a one-to-one correspondence between C and all points on S2 except the North pole, Figure 11.3. It is easy to verify that (ξ , η, ζ ) corresponds to

·

ξ ¸ +i· η ¸ 1−ζ 1−ζ

As ( ξ , η, ζ ) gets near to (0, 0, 1) it follows (as is obvious geometrically) that | x + iy| becomes very large. Thus it is reasonable to introduce the symbol

Figure 11.3 The Riemann sphere.

236

Laurent Series

Figure 11.4 Neighbourhood of

∞ on the Riemann sphere.

to correspond to (0, 0, 1) ∈ S2 . Thus we have a one-to-one correspondence between S2 and C ∪ {∞}. We call the latter the extended complex plane. It may be identiﬁed with S2, which is then known as the Riemann sphere. We can now think of C ∪ {∞} either as a plane plus an extra point, or as a sphere. Correspondingly, we think of C as a plane, or as a sphere without a North pole. Both viewpoints are valuable, depending on the problem at hand. Since {x + iy : | x + iy| > R } corresponds to a ‘spherical cap’ between a line of latitude and the North pole (Figure 11.4) it makes sense to think of {z ∈ C : |z| > R} as a ‘neighbourhood of ∞ ’. Such neighbourhoods get smaller as R gets bigger. So doing this leads to a concept of continuity on S2, agreeing with geometric intuition (and the standard distance function on R3 ). Readers familiar with topology can render this in more precise terms: we obtain a topology on the Riemann sphere that is identical to its usual topology as a subset of R3 .

11.5

Behaviour of a Differentiable Function at Inﬁnity Suppose that f (z) is differentiable in {z ∈ C : |z|

> R}. Then we can deﬁne g(z) = f (1/z) (0 < |z| < 1/R) Since g² (z) = − z−2f ² (1/z) it follows that g(z) is differentiable for 0 < |z| < 1/R, and therefore has an isolated singularity at 0.

11.14. The function f has a removable singularity, pole of order m, or isolated essential singularity at ∞ if and only if g(z) = f (1/z) has the corresponding type of singularity at 0.

D E FI N I T I O N

11.6 Meromorphic Functions

Example 11.15. f (z) singularity at .

Example 11.16. f (z) Example 11.17. f (z) singularity at .

= 1/z (|z| > 0). Then g(z) = z (|z| > 0), so f

237

has a removable

= z. Then g(z) = 1/z ( |z| > 0), so f has a simple pole at ∞. = ez. Then g(z) = = /

e1/ z (| z|

±= π ∈

>

0) and f has an isolated essential

= /

/

Example 11.18. f (z) 1 (sin z) (z n , n Z). Then g(z) 1 sin(1 z). We cannot say that f has an isolated essential singularity at , because f is not differentiable in z C: z R for any R 0. However, f certainly has some sort of singularity at . In general, if f (z) has a sequence of isolated singularities zn such that zn , we say that f has an essential (but not isolated) singularity at .

{ ∈

| |> }

>

→∞

If f has a removable singularity at ∞ then g has a removable singularity at 0, so may be rendered differentiable in {z ∈ C : |z| < 1/R }. Thus we may say that f is differentiable (or analytic) at ∞, and deﬁne f (∞)

and now

= g(0) = zlim g(z) →0

f (∞ ) = lim f (z) z →∞

For instance, if f (z) = 1/z then g(0) = 0 so f (∞) = 0. Further, we say that f has a zero of order m at ∞ if g has a zero of order m at 0. Thus if f (z) = 1/zm (|z| > 0, m ∈ Z, m > 0) then g(z) = zm and f has a zero of order m at ∞ . In general, any statement about the behaviour of f at ∞ can be translated into one about g at 0 – and this is how to deﬁne or prove such a statement.

11.6

Meromorphic Functions Poles are not, as singularities go, particularly nasty, and it is of interest to consider a class of functions more general than differentiable ones. D E FI N I T I O N 11.19. If f is differentiable everywhere in a domain D except for points at which f has a pole, then f is meromorphic in D.

For instance, f (z) = 1/z is differentiable in C \ {0}, and has a pole at 0, so it is meromorphic in C. Similarly, f (z) = 1/(sin z) is differentiable in C \ { nπ : n ∈ Z}, and has poles at nπ , so it is meromorphic in C. We might go further, and consider functions meromorphic in the extended complex plane – by which we mean that the only singularities of f , including ∞ , if necessary are

238

Laurent Series

poles. It is not necessary to introduce a special term for such functions, because they turn out to have a simple description. Recall that a rational function is one of the form f (z)

= φ(z)/ψ (z)

where φ , ψ are polynomial functions. Then we have: 11.20. A function is meromorphic in the extended complex plane if and only if it is rational.

P RO P O S I T I O N

Proof. Clearly a rational function is meromorphic in C ∪ {∞} . Suppose conversely that f is meromorphic in C ∪ {∞}. Since f is differentiable at ∞ or has a pole at ∞ , there exists R > 0 such that f is differentiable in

{z ∈ C : |z| > R} So the poles, other than ∞ , occur inside the closed disc

{z ∈ C : |z| ≤ R} Since poles are isolated singularities, each pole has a neighbourhood that contains no other poles. Indeed, if there were an inﬁnite set S of poles inside |z| ≤ R, there would be a limit point of S. This is not possible because the poles are isolated. This closed disc contains only ﬁnitely many poles, say z1 , . . . , zk . Let the orders of these poles be n1, . . . , nk . Deﬁne g(z) = (z − z1 )n1 · · · (z − zk )nk f (z)

This is differentiable throughout C, hence analytic, so it has a Taylor series

∞ ±

g(z) =

n=0

(z ∈ C)

an zn

Now the polynomial

ψ (z) = (z − z1)n · · · (z − zk )n has a pole of order n1 + · · · + nk at ∞, since 1

ψ

· ¸ 1 z

=

·

1 z

− z1

¸n1

k

···

·

1 z

− zk

¸nk

1 + ··· = zn +···+ n Since f has at worst a pole of order N at ∞ for some N it follows that g(z) has a pole of 1

order

k

M = n1 + · · · + nk

at ∞. Now g(1/z) =

∞ ± n=0

an z−n

+N

(| z| > 0)

11.7 Exercises

and since this has a pole of order M at 0 we must have an

239

= 0 for n > M. Therefore

= a0 + · · · + aMzM which is a polynomial, so f (z) = g(z) /ψ (z), which is rational. g(z)

A function meromorphic only in C need not be rational, as the example 1/(sin z) shows. This is not rational since it has inﬁnitely many poles, but we have already seen that it is meromorphic in C. So the behaviour at inﬁnity is important.

11.7

Exercises 1. Find the Laurent expansions of the following functions of z around z = 0: (i) (z − 3)−1 (ii) (z − a)−k (a ∈ C, k = 1, 2, 3, . . .) (iii) 1/ (z(1 − z)) (iv) 1/ ((z − a)(z − b)) (a, b, ∈ C) (v) z3e1/z (vi) exp(z + 1/z) (vii) cos(1/z) (viii) exp(z−5) In each case, specify the largest annulus in which the expansion is valid. 2. Find the Laurent expansions of the following functions on the stated annulus: (i) (z − 1)−2(z − 2)−1 on 0 < |z| < 1 (ii) (z − 1)−2(z − 2)−1 on 1 < |z| < 2 (iii) (z − 1)−2(z − 2)−1 on 2 < |z| < 3 (iv) exp(−z−2 ) on | z| > 0 (v) (1 − z − z2 )−1 in powers of z − 1 on 0 < |z − 1| < 1 (vi) ez / (1 + z2) on | z| > 1 3. Which of the following functions have a Laurent expansion around the given point z0 (that is, in powers of z − z0 )? (For multivalued functions, choose one particular deﬁnition on a domain that makes it single-valued.) √ (i) z about z0 = 1 √ (ii) z about z0 = 0 (iii) Log z about z0 = 0

(iv) Log z about z0 = 3

√z about z = 0 0 − 1 (vi) tan (1 + z) about z0 = 0 (vii) sin−1(z) about z0 = 0 ¼ (viii) (π/2) − sin−1 (z) about z0 = 1 (ix) z2 cosec(1/z) about z0 = 0 ¼ √ (x) (π/4) − sin−1 (z) about z0 = 1/ 2 (v)

»

1+

240

Laurent Series

4. Prove the validity of the Laurent expansion 1 (z − 1)(z − 2)

=

∞ ∞ ± ± 2−(n+1)zn + z−n n=0

n=1

on a suitable annulus, and state which annulus. 5. Find Laurent series for (z2 − 1)−1 and (z2 + 1)−1 in powers of z + i and z − i, and say in which annuli these are valid. 6. Let a, b ∈ C. Show that

exp(az + bz −1) = where an

= 21π

´ 2π 0

−∞

an zn

e(a+b) cos θ cos[(a − b) sin θ

7. Let a ∈ C. Show that sin(a(z + z−1)) where an

∞ ±

= 21π

´ 2π 0

=

∞ ± −∞

− nθ ]dθ

an zn

sin(2a cos θ ) cos nθ dθ

8. Find the poles of the functions: (i) 1/(z2 + 1) (ii) 1/(z4 + 16) (iii) 1 /(z4 + 2z2 + 1) (iv) 1/(z2 + z − 1) 9. Describe the type of singularity at 0 of each of the following functions: (i) sin(1 /z) (z ± = 0) (ii) z−3 sin2 z (iii) z cot z (z ± = nπ , n ∈ Z) (iv) cosec2 z − z−2 (z ± = nπ , n ∈ Z) (v) (cos z − 1)/ z2 (vi) (sin z − z + z3/6)/z7 10. Let D be a disc centre z0 , let f be differentiable on D except at z0, and suppose that | f (z)| is bounded on D \ {z0}. Show that z0 is a removable singularity of f . (Hint: negative Laurent terms are 0 – why?) 11. Find all singularities of the following functions, and say which are poles: (i) (z + z−1)−1 cos π z (ii) 1 − 4z2 (iii) exp(z + z−1 )

11.7 Exercises

241

12. Construct a function deﬁned on C ∪ {∞} having only the following singularities: (i) A pole of order 2 at ∞. (ii) A simple pole at each of the points e2π ik/p , k = 0 , . . . , p − 1, p an odd integer ≥ 2. √ (iii) A simple pole at z = 2 and a pole of order 5 at z = 2. 13. Let p(z), q(z) be polynomials of degrees m, n respectively. Describe the behaviour at inﬁnity of: (i) p(z) + q(z) (ii) p(z)q(z) (iii) p(z)/q(z) 14. Find all singularities, and the behaviour at inﬁnity, of the functions: (i) (z − z3 )−1 (ii) z5/(2 − z2)2 (iii) (e z − 1)−1 − 1/z (iv) cot 1/z (v) (cos z)z−2 (vi) ((cos z) − 1)z−2 (vii) cot(1/z) − 1/z (viii) sin(1/ cos(1/z))

15. Let ± be a circle in the complex plane, or a straight line. Is its image on the Riemann sphere also a circle? 16. Find the poles and zeros of tan z. Show that tan z is meromorphic in C, but is not a rational function. 17. Show that (z + 1 + z−1 )−1 has a removable singularity at z expansion, and the radius of convergence of this.

= 0. Find its Taylor

18. Verify Picard’s Theorem directly for the functions: (i) ez (ii) tan2 z (iii) z2 (iv) sin z (v) e1/ z (vi) cos z (vii) tan z (viii) A function of your own choice.

19. Let f (z) have a pole of order n at z = a. Deﬁne the principal part φ(z) of f (z) to be the sum of the negative-power terms in the Laurent expansion of f (z) in powers of z − a. Prove that f (z) − φ (z) is differentiable at z = a. 20. Show that in a neighbourhood of a pole, a complex function is the sum of a rational function and a differentiable one.

21. Suppose that f is differentiable on C except at poles, and that ∞ is either a pole or a removable singularity of f . Show that:

242

Laurent Series

(i) f has only ﬁnitely many poles. ∑ (ii) f − r pr is constant, where the poles of f are at points ar and the pr are the corresponding principal parts of f (deﬁned in Exercise 19). ∑ (iii) f is a rational function, and r pr is its ‘partial fraction’ decomposition.

22. Let f : C → C be differentiable, with f (z) ± = 0 for all z limz →∞ f (z) exists and is non-zero. Prove that f is constant.

∈ C. Suppose that

12

Residues

Among the many applications of complex analysis are the explicit computation of definite integrals, the summation of series, and counting how many zeros a function has in a given region of C. Although such questions are not as important a part of pure mathematics as they once were, they are still very relevant to practical applications. Further, the power of the method and its wide applicability demonstrate the advantage of general principles and deep theorems over manipulative ingenuity. The basic idea in all of these applications is to use Cauchy’s Theorem to exploit the exceptional nature of the term b1/(z − z0 ) in the Laurent expansion of a differentiable function.

12.1

Cauchy’s Residue Theorem The residue of f at z0 is determined by the special coefﬁcient b1 just described: D E FI N I T I O N 12.1.

If f has an isolated singularity at z0 and a Laurent expansion

f (z0 + h) = then the residue of f at z0 is

∞ ± n=0

a nh

n

+

∞ ± n=1

bnh−n

(0 < |h| < R)

res ( f , z0 ) = b1

From Theorem 11.1 we immediately deduce that res ( f , z0 ) = where Cr (t) = z0 + eit (t to integration.

1 2π i

² Cr

f (z)dz

(12.1)

∈ [0, 2π ]) and 0 < r < R. This shows the relevance of residues

D E FI N I T I O N 12.2. A closed path γ is a simple loop if every point z not on γ has w(γ , z) = 0 or w(γ , z) = 1. As usual, the set of points z ∈ C satisfying w(γ , z) ± = 0 (which in this case means w(γ , z) = 1) are said to be inside γ .

In applications, simple loops will all be made up of straight line segments and parts of circles, as in Figure 12.1. All the simple loops encountered will be Jordan contours (that

244

Residues

Figure 12.1 Simple loops in

C.

Figure 12.2 A simple loop in

C that is not a Jordan contour.

is, they do not self-intersect), but that is not essential for the theory. (Figure 12.2 shows a simple loop according to our deﬁnition that is not a Jordan contour.) What matters is that the points inside γ all have w(γ , z) = 1. (In particular, γ winds once anticlockwise around such points.) In such cases, we have: 12.3 (Cauchy’s Residue Theorem). Let D be a domain containing a simple loop γ and the points inside γ . If f is differentiable in D except for ﬁnitely many isolated singularities at z 1, . . . , zn inside γ , then

THEOREM

²

γ

f (z)dz = 2π i

n ± r =1

res ( f , zr )

Proof. Since D is open we can ﬁnd circles Sr (t) = zr + εr eit (t ∈ [0, 2π ]) round the zr such that Sr and points inside it lie inside D, and such that Sr contains no singularity of f other than zr , Figure 12.3. Then the collection of paths

−γ , S1 , . . . , Sn satisﬁes the hypotheses of Theorem 8.9, because w(−γ , z) w(−γ , zr ) w(Sm , zr )

= w(Sr , z) = 0 = −w(γ , zr) = −1 = 0 if m ±= r, 1

(z ± ∈ D) since zr is inside γ if m = r

12.1 Cauchy’s Residue Theorem

245

Figure 12.3 Small circular contours around each isolated singularity.

Therefore

w(−γ , z) + w(S1 , z) + · · · + w(Sn, z) = 0

for all z ± ∈ D. By the Generalised Cauchy Theorem,

²

−γ so that

²

γ

f

+

²

S1

f

²

f

+ ··· +

² Sn

f

=0

²

= f + ··· + f S S = 2π i · res ( f , z1) + · · · + 2π i · res ( f , zn ) 1

n

by (12.1). Alternative Proof. Instead of the Generalised Cauchy Theorem we can use Laurent series: the proof is instructive but less elegant. Around each zr there is a Laurent expansion of f , which we split into three parts: f (z) = Qr (z) + where Qr (z) = Pr (z) =

res ( f , zr ) z − zr

∞ ± n=2

∞ ±

n=0

+ Pr(z)

(12.2)

bn (z − zr )−n an (z − zr )n

Now Pr (z) is differentiable in a neighbourhood of zr , and Qr (z) + res ( f , zr )/ (z − zr ) is differentiable for z ± = zr since all but a ﬁnite number of the bn are zero. Therefore h(z) = f (z) −

n ± r =1

Qr (z) −

n ± res ( f , zr ) j =1

z − zr

(12.3)

246

Residues

is differentiable in D, except perhaps at z1, . . . , zn. But from (12.2) it is also differentiable in a neighbourhood of each zr . Hence h(z) is differentiable in D, so by Cauchy’s Theorem

²

But Qr (z)

γ

= Tr² (z) where Tr (z) =

∞ ± n=2

so that

²

γ

h(z)dz = 0

(12.4)

bn −n+1 −n + 1 (z − zr ) Qr (z)dz = 0

(12.5)

as well. Integrating round γ and using (12.4) and (12.5), we get 0=

= 12.2

²

γ

²

γ

f (z)dz − 0 − f (z)dz − 2π i

n ±

r =1 n ± r =1

²

res ( f , zr )

γ

(z − zr )−1 dz

res ( f , zr )

Calculating Residues The Residue Theorem can be used to calculate integrals – and not just round simple loops. For it to be of much use, we must ﬁnd ways to calculate residues. The following two lemmas are very useful in this respect. LEMMA

12.4. (i) If z0 is a simple pole of f then

= zlim →z (z − z0 )f (z) If f (z) = p(z)/q(z) where p(z0) ± = 0, q(z0) = 0, q² (z0) ± = 0, then res ( f , z0) = p(z0 )/q² (z0 ) res ( f , z0)

(ii)

0

Proof. We have already done part (i) in the previous chapter as Proposition 11.10, but to recap: we have f (z)

= z −b1z + 0

and so (z − z0)f (z) = b1 + which tends to b1 as z → z0 . For (ii), note that

(z − z0 )p(z) lim z →z 0 q(z)

since q(z0)

∞ ± n=0

an(z − z0)n

∞ ± n=0

= zlim p(z) →z 0

= 0, and this is equal to p(z0 )/q² (z0).

an(z − z0)n+1

³´

q(z) − q(z0 ) z − z0

µ

12.2 Calculating Residues

For example, if f (z) then res ( f , 1) LEMMA

πz = cos z976

1 = 976cos· 1π975 = 976

12.5. If z0 is a pole of f of order m then res ( f , z0) = lim

z →z0

·

dm−1 ((z − z0 )m f (z)) (m − 1)! dzm−1 1

Proof. We have f (z) =

bm (z − z0)m

+ · · · + z −b1z + 0

∞ ± n=0

so that (z − z0 )m f (z) = bm + · · · + b1 (z − z0 )m−1 + Therefore

dm−1 ((z − z0 )m f (z)) dzm−1

247

= (m − 1)! b1 +

an (z − z0 )n

∞ ± n=0

an (z − z0 )m+n

∞ (m + n)! ± n=0

(n + 1)!

an (z − z0 )n+1

Now take limits. For example, consider f (z) =

´

which has a triple pole at z0 = 1. Then

z+ 1 z− 1

µ3

(z − 1)3f (z) = (z + 1)3

and so

6 1 d2 ((z − 1)3 f (z)) = (z + 1) 2 2! dz 2!

which tends to 3 · 2 = 6 as z → 1. So res ( f , 1) = 6. On occasion another technique may be brought into play: working out the appropriate part of the Laurent series. (It is a waste to work out the whole thing, because the whole point about residues is that we do not need the whole thing, but only b1 .) For instance: 1 f (z) = 2 z sin z µµ ´ ´ z3 2 = 1/ z z − 6 + · · ·

=

1 z3

´

1−

z6 6

+ ···

µ−1

248

Residues

´

= z13 so that res ( f , 0)

12.3

1+

z6 6

+ ···

µ

= z13 + 6z1 + · · ·

= 1/6.

Evaluation of Deﬁnite Integrals We now consider a number of techniques for calculating various kinds of deﬁnite integral.

I:

±

Let C(t)

0

Q(cos t, sint)dt

= eit (t ∈ [0, 2π ]) be the unit circle. If z = C(t) = eit

then cos t = sin t = from which we get

² 2π 0

²

´

´ ´

where ± is the sum of the residues of

´ ´

µ

µ

´

1 1 1 1 1 Q z+ , z− iz 2 z 2i z inside C. For example, consider

² 2π 0

Then (12.6) becomes 1 iz

¸ ´

1 1 z+ 8 z

µ3

´

1 1 1 1 z+ , z− 2 z 2i z

= Q C = 2π i±

Q(cos t, sin t)dt

µ

1 1 z+ 2 z ´ µ 1 1 z− 2i z

µµ

dz iz

µµ

(cos3 t + sin2 t)dt

1 2 1 3 z − z+ 8i 4i 8i

´

1 1 z− 4 z

µ2¹

=

1 + 2iz + 8iz3 2 − 4iz1 3 + 8iz1 4 which has a pole inside C with residue 1/ 2i. So the integral is 2π i/2i = π .

(12.6)

12.3 Evaluation of Deﬁnite Integrals

249

If Q is at all complicated, the computations can become very tedious. Sometimes integrals of this kind can be found from the real and imaginary parts of an integral

²

C

g(z)dz

with a suitable choice of g. For instance,

²

C

ez dz = 2π i z

since e /z has residue 1 at z = 0. Therefore z

² 2π cos t+i sin t e eit

0

so

² 2π 0

and

² 2π 0

dt = 2π i

ecos t +i sin t dt

= 2π

ecos t [cos(sin t)) + i sin(sin t)]dt = 2π

Equating real and imaginary parts,

² 2π 0

² 2π 0

II:

± ∞

−∞

ecos t cos(sin t) = 2π ecos t sin(sin t) = 0

f (x)dx

The real integral

² ∞

−∞

f (x)dx

is deﬁned to be

²

lim

x2

x1 →−∞ , x 2 →∞ x 1

(12.7)

f (x)dx

(12.8)

provided the limit exists. The techniques we are about to discuss allow the calculation of lim

²

R→∞

R

−R

f (x)dx

which is known as the Cauchy principal value of the integral, denoted by P

² ∞

−∞

f (x)dx

(12.9)

250

Residues

If (12.8) exists then so does (12.9) and the two are equal. But the Cauchy principal value may exist when (12.8) does not. For example

²

R

−R

x dx = [x2/ 2]R−R

so that

² P

R

=0

x dx = 0

−R

But clearly (12.8) does not exist when f (x) = x. It follows that when we use this technique below, we must take into account the convergence of (12.8). This, in part, leads to condition (ii) of the following: P RO P O S I T I O N

12.6. Suppose that:

(i) A function f is differentiable in the upper half-plane {z ∈ C : im z ≥ 0} except for a ﬁnite number of poles, none of which lies on the real axis. (ii) If SR (t) = Reit (t ∈ [0, 2π ]) then there is a constant A such that for large R

|f (z)| ≤ A/R2

when z lies on SR . Then

² ∞

−∞

f (x) dx = 2π i ±

where ± is the sum of the residues of the poles of f in the upper half-plane. Proof. Choose R large enough for (ii) to hold, and so that all the poles lie inside SR + [−R, R], Figure 12.4. By Cauchy’s Theorem,

²

R

−R

f (x) dx +

with ± as stated. Now let R → ∞ . Then

²

SR

f (z)dz = 2π i ±

º º² º º º f (z)dzº ≤ A π R = π A º R2 º S R R

Figure 12.4 Choice of R in proof of Proposition 12.6.

12.3 Evaluation of Deﬁnite Integrals

which tends to 0 as R → ∞ . Hence

²

lim

R→∞

That is, P

R

−R

² ∞

−∞

251

f (x) dx = 2π i ±

f (x) dx = 2π i ±

Now, (ii) tells us that |f (x) | ≤ A/x2 for large |x|, so it follows from real analysis that

² ∞

−∞

f (x) dx

exists. It is therefore equal to its Cauchy principal value, hence to 2 π i ±. Condition (ii) is certainly satisﬁed if f (z) = p(z)/q(z) where p and q are polynomials such that q has no real zeros and the degree of q is greater than or equal to the degree of p plus 2. R E M A R K 12.7.

For example, consider

² ∞

dx 2 + a2 )(x2 + b 2) (x −∞

where a, b > 0 are real, a ± = b. By Remark 12.7 this satisﬁes (ii), and (i) is obviously true. The only poles of 1/((z2 + a2 )(z2 + b2 )) in the upper half-plane are simple poles at ia, ib. The residue at ia is lim

z →ia (z2

z − ia + a2 )(z2 + b2)

= 2ia(b21− a2 )

and similarly that at ib is 1 2ib(a2 − b2) Therefore the value of the integral is 2π i

´

1 2ia(b2

µ

+ 2ib(a2 − b2) = ab(aπ+ b) 1

a2 )

12.8. In the proof of Proposition 12.6 we do not require f (z) to be real for z on the real axis. REMARK

For instance, let f (z) =

+

eiz

+ b2 ) Then on SR we have | eiz | = |e−y+ix | = e−y ≤ 1 for y ≥ 0, so (ii) holds. As before, there (z2

a2)(z2

are simple poles at ia, ib, but now the residues are e−a 2ia(b2 − a2 )

and

e−b 2ib(a2 − b2)

252

Residues

Hence

² ∞

eix dx = π −∞ (x2 + a2 )(x2 + b2)

´

e−a

a(b2 − a2 )

e−b

+ b(a2 − b2 )

µ

so, equating real and imaginary parts,

² ∞

cos x π dx = 2 2 + a 2)(x2 + b 2) (x b − a2 ²−∞ ∞ sin x dx = 0 2 −∞ (x + a2)(x2 + b2)

´ −a e a

−b µ

− eb

Of these, the second equality is actually obvious (why?), but the ﬁrst is not. We » can try to generalise this method in (at least) two ways: by getting a better estimate for SR f or by allowing f to have poles on the real axis. The ﬁrst is dealt with in (III), the second in (IV).

III:

±∞

−∞

P RO P O S I T I O N

f (x)eix dx 12.9. Suppose that:

(i) A function f is differentiable in a domain containing the upper half-plane {z im z ≥ 0} except for a ﬁnite number of poles, none on the real axis. (ii) There is a constant A such that for large R

|f (z)| ≤ A/ R when z lies on SR . Then

² ∞

−∞

∈C:

for |z| = R

f (x)eix dx

= 2π i ±

where ± is the sum of the residues of the poles of f (z)eiz in the upper half-plane. Proof. It is possible to use the same contour as in (II) and prove that

²

lim

R→∞ SR

f (z)eiz dz = 0

but this calculates only the Cauchy principal value. Moreover, the convergence problem in this case is much harder because all we know is that f behaves like 1/|x| for large | x| , which on its own does not imply the existence of

² ∞

−∞

f (x)eix dx

More delicate arguments can overcome this obstacle, but it is easier to sidestep the whole question by using a different contour, as in Figure 12.5.

12.3 Evaluation of Deﬁnite Integrals

Figure 12.5 Contour for the proof of Proposition 12.9.

We prove that as X 1, X 2, Y → ∞ , each of

²

²r

f (z)eiz dz

for r = 2, 3, 4 tends to zero. It then follows as before that

²

lim

X 1 ,X2 →∞

X2

−X1

f (x)eix dx = 2π i ±

as required. Now we carry out the necessary estimates,

º² º º² Y º º º º º iz iX − t 2 º f (z)e dzº = º idt ºº º ² º º 0 f (X2 + it)e 2 ² Y

≤ for X2 large. Similarly

0

A X2

A −t e dt X2

º² º º º º f (z)eiz dzº ≤ A º ² º X1 4

Also

º² º º ² X2 º º º º º it −Y º º f (z)eiz dzº = º− dt º º ² º º −X f (t + iY)e 3 º º² X2 1 º º ≤ ºº YA e−Y dt ºº

−X1 = AY −1e−Y (X1 + X2 )

For ﬁxed X1 , X2 this tends to 0 as Y

→ ∞. Now let X1 , X2 → ∞.

As an example, take

² ∞

x3 eix dx −∞ (x2 + a2)(x2 + b2)

(a, b > 0, a ± = b)

253

254

Residues

which does not satisfy the conditions for (II), but does for (III). The integrand has simple poles in the upper half-plane at ia and ib. Calculating the residues in the usual way, applying Proposition 12.9, and equating imaginary parts yields

² ∞

π (b2e−b − a2e−a) x3 sin x dx = 2 2 + a 2)(x2 + b 2 ) (x b − a2 −∞

(Equating real parts proves that the corresponding cosine integral is zero, but again this is obvious on other grounds.)

IV:

Poles on the real axis.

If f (z) has poles on the real axis we make ‘indentations’ in the contour by drawing small semicircles as in Figure 12.6. Suppose these semicircles have radii ε1 , ε2, . . .. Then we proceed as above, but letting ε1, ε2 , . . . tend to 0 at the same time as R, X1 , X 2, Y → 0. There is a problem here similar to that in case (II): all we calculate is a Cauchy principal value

² P

b

a

f (x)dx = lim

¸²

ε→0

x0 −ε

a

f (x)dx +

²

x0 +ε

for a pole at x0. This again may exist even if

²

a

b

f (x)dx = lim

ε 1 →0

²

x0 −ε

f (x)dx + lim

ε2→0

a

does not. Thus

²

¹

b

²

f (x)dx

b

x 0 +ε

f (x)dx

1

dx −1 x

does not exist, but

² P

1

dx −1 x

=0

Consequently there is a convergence problem, once the Cauchy principal value has been obtained. Apart from this consideration, the hypotheses and conclusions of Propositions 12.6 for case (II) and 12.9 for case (III) remain valid even when poles occur on the real axis, provided that ± is summed only over the non-real poles in the upper half-plane.

Figure 12.6 Types of contour used when there are poles on the real axis.

12.3 Evaluation of Deﬁnite Integrals

255

Figure 12.7 Contour used when there are poles on the real axis.

Rather than stating a cumbersome general theorem, we content ourselves with a typical example:

² ∞ ix e

−∞ x

dx

There is a real pole at 0, so we take a contour as in Figure 12.7. As before, the integrals along ²2, ²3 , and ²4 tend to zero. Since there are no poles inside the contour, lim

X1 ,X2 →∞

¶² −ε ix e

−X1 x

But eiz z

=

dx +

²

X2

ε

eix dx + x

1 ± inzn−1 + z n! n=1

²

eiz dz Cε z

·

=0

= 1z + φ(z)

where φ is differentiable. Hence |φ(z)| ≤ M in a neighbourhood of 0, so

² ´

²

µ

eiz 1 dz = lim + φ(z) dz lim ε→0 Cε z ε→0 Cε z ´ ² π µ ² 1 it φ (z)dz = ε→ lim − i ε e dt + lim it 0 ε→0 Cε 0 εe = iπ using the Estimation Lemma. Therefore P Equating real and imaginary parts, P

² ∞ ix e

−∞ x

dx = iπ

² ∞ cos x

dx = 0

x ²−∞ ∞ sin x P dx = π −∞ x

256

Residues

The ﬁrst integral exists only as a principal value, since cos(x)/x behaves like 1 /x for small |x|. But for the second we have P

² ∞ sin x

−∞ x

dx = lim

ε→0

´² −ε sin x

= 2 ε→ lim 0

x ²−∞ ∞ sin x x

ε

dx +

µ ² ∞ sin x

ε

x

dx

dx

This limit exists, hence by deﬁnition it equals 2

² ∞ sin x x

0

dx

because sin(x)/x → 1 as x → 0. Hence we can remove the P from the second expression above. Further, we have also proved that

² ∞ sin x

V:

±∞

−∞

x

0

dx =

π

2

eax dx φ (ex)

For integrals of this type we integrate around a contour like that in Figure 12.8. In this case, on ²3 , if we substitute z = −t + 2π i, then −X 2 ≤ t ≤ X1, so that

²

²

X1 e−at+2π ia eaz dz = (−1)dt −X2 φ (e−t +2π i ) ² 3 φ (ez) ² −X 2 −at e = e2π ia dt φ (e−t ) X1

Putting t

= −x, this equals

−e2π ia If φ is such that

²

eaz dz → 0 ²r φ(ez)

then we obtain (1 − e

2π ai

²

as

X2

−X1

eax φ (ex ) dx

X1, X2

² ∞ ax e

−∞ φ (ex)

Figure 12.8 Contour for integrals of type (V).

→ ∞ (r = 2, 4)

dx = 2π±

12.3 Evaluation of Deﬁnite Integrals

where ± is the sum of the residues of eaz/φ (z) at poles between the lines im z im z = 2π . For example, consider

² ∞

eax

−∞ e2x + 1

dx

257

=

0,

(0 < a < 1)

The relevant singularities are at iπ/2, 3iπ/2, with corresponding residues

− 21 e3iπ a/2

− 21 ei π a/2

Hence the value of the integral is

2π i (− 1 eiπ a/2 − 21 e3iπ a/2) 1 − e2π ia 2

Putting k = eiπ a/ 2 this is easily seen to be equal to

π 2 sin(π a/2)

VI:

Short cuts.

You should always be on the lookout for quick methods, other than those given above. We give two examples, each of independent interest.

Example 12.10. We use contour integration to prove that the integral of the ‘bell curve’ or ‘gaussian’, which is of importance in probability theory, is

² ∞

−∞

e−x dx = 2

π

This result is usually obtained by squaring the integral, rewriting it as

´² ∞

−∞

2 e−x dx

µ ´² ∞

−∞

2 e−y dy

µ

=

² ∞² ∞

−∞ −∞

2 2 e−(x +y )dxdy

and then changing to polar coordinates. For a long time it was thought that no evaluation by contour integration existed, but several such proofs are now known. We give one of them, from Remmert [15]. Consider f (z) =

e−z 1 + e−2az

with a = (1 + i) Then a2

= iπ , so

2

¼

π

2

= e−z (12.10) The poles of f are simple and located at the points − 21 a + na with n ∈ Z. Now integrate f over the rectangle with corners −r, s, s + i im(a), −r + i im(a) with r, s > 0. The only pole of f inside this rectangle is at 2a , and its residue is f (z) − f (z + a)

2

258

Residues

res( f , a/2) =

e− 4 a

1 2

−2ae−a2

= 2−√iπ

This follows from Lemma 12.4, which states that res

½g ¾ h

,x

= hg(x) ² (x)

if h has a simple zero at x and g(x) ± = 0. Using the residue theorem and (12.10), it is easy to see that

² ∞

−∞

²

e−x dx = lim 2

r,s →∞

s

−r

e−x dx 2

= 2π i res( f , a/ 2) = √π

since the integrals along the vertical sides of the rectangles converge to zero as r, s → ∞. Example 12.11. An integral of great importance in applied mathematics is

² ∞

−∞

ei λx e−x dx 2

The function e−z is differentiable throughout −R, R, R − iλ/2, −R − iλ/2, then 2

²

e−z dz 2

γ

(λ ∈ R)

C. If γ

is the rectangle with vertices at

=0

As R → ∞ , the integrals over the vertical edges of the rectangle tend to zero. The other two converge. Hence

² ∞

−∞ Therefore

² ∞

−∞

e−x dx = 2

= i λx

² ∞

²−∞ ∞

−∞

e−(x −iλ/2) dx 2

2 2 e−x eiλx eλ /4 dx

2 2 e e−x dx = eλ /4

² ∞

−∞

e− x dx 2

The latter integral is a constant; it has just been evaluated in Example 12.10 and equals √π . So

² ∞

−∞

2 2 √ eiλx e−x dx = eλ / 4 π

(Although we can derive Example 12.10 from this by setting ple 12.10 to get the result.)

12.4

λ = 0, we need Exam-

Summation of Series Residues can also be used to ﬁnd explicit expressions for sums of suitable inﬁnite series, based on the idea that the functions cot π z and cosec π z have simple poles at

12.4 Summation of Series

z=n∈ that

259

Z . If f is a function that is differentiable at all z = n ∈ Z then it is easy to check res ( f (z) cot π z, n) = res ( f (z) cosec π z, n) =

f (n)

π

(−1)n f (n)

π

This suggests a method for summing certain series, as follows. Let CN be the square whose vertices are (N

+ 21 )(³1 ³ i)

parametrised, as usual, in the anticlockwise direction, Figure 12.9. We claim that cot π z and cosec π z are bounded on CN , where the bound is independent of N. To prove this, ﬁrst note that on the two horizontal sides z = x + iy, where |y| ≥ 21 . Now

º iπ z º º e + e−iπ z º º | cot π z| = º eiπ z − e−iπ z ºº º iπ z º º | e | + |e−iπ z | º º ≤ º |eiπ z| − |e−iπ z| ºº º −π y º º eπ y ºº = ºº ee−π y + − eπ y º

= coth |π y| ≤ coth π/2

Also

|cosec π z| = ( 21 |ei π z − e−i π z|)−1 ≤ ( 12 |e−π y − eπ y |)−1 = (sinh |π y|)−1 ≤ (sinh π/2)−1

Figure 12.9 Square contour used to sum series.

260

Residues

+ 21 ) + it, so | cot π z| = º| tanh π t| º −2π t º º = ºº 11 +− ee−2π t ºº ≤1

On the other two sides of the square, z = ³(N

and

|cosec π z| = | sin π z|−1 = | cos(iπ t) |−1 = (cosh |π t |)−1 ≤1 Hence there is a constant M such that | cot π z| ≤ M and |cosecπ z| ≤ M for any z on C N . Suppose now that for large enough |z| we have |f (z)| ≤ |zA|2 Then we claim that

±=0 where ± is the sum of the residues of f (z) cot π z. By Cauchy’s Residue Theorem, ² f (z) cot π z dz = 2π i ±N CN

where ±N is the sum of the residues of f (z) cot π z inside CN . As N ±, so it is sufﬁcient to prove that the integral tends to 0. But

→ 0, clearly ±N →

º² º º º A º º º C f (z) cot π z dzº ≤ N 2 M(8N + 4) N

for large enough N by the Estimation Lemma. As N → 0, this tends to 0 as claimed. Usually, ± is an inﬁnite series, and since it is zero we can sum certain related series. This is best illustrated by an example, and the obvious one to try is f (z) = 1/z2. At an integer n ± = 0 the function z−2 cot π z has a simple pole with residue 1/(n2π ), whereas at the origin it has a triple pole with residue −π/ 3. Hence 0=±

= − π3 +

−1 ±

n=−∞

1

+ n2 π

∞ ± = − π3 + π2 n12

∞ 1 ± n=1

n2 π

n=1

Hence

∞ 1 ± n=1

π = 2 n 6

2

which is a theorem originally proved by Euler, using a different (non-rigorous) method.

12.5 Counting Zeros

261

If we use cosec π z instead of cot π z a similar result applies, allowing us to sum series ∑ of the form (−1)n f (n). For instance, using f (z) = 1/z2 and arguing much as above, we can prove that

∞ ± n=1

12.5

(−1)n+1

1 n2

= π12 2

Counting Zeros A rather different use for residues is to calculate the number of zeros of a function that is differentiable inside a simple loop. We begin with: T H E O R E M 12.12. Suppose that f is differentiable, apart from a ﬁnite set of poles, in a domain D containing a simple loop γ and all points inside γ . If f has no zeros or poles on γ then ² ² 1 f (z) dz = N − P 2π i γ f (z)

where N is the number of zeros of f inside γ and P is the number of poles of f inside γ , each counted according to multiplicity.

Proof. By Cauchy’s Residue Theorem, the integral is the sum of the residues of f ² (z)/f (z) at its poles inside γ . If z0 is neither a zero nor a pole of f , then f ²/ f is differentiable at z0 . We show that: (i) If f has a zero of order k at z1 then f ² /f has a pole with residue k. (ii) If f has a pole of order m at z2 then f ² /f has a pole with residue −m. To prove (i) note that in that case where φ (z1)

f (z) = (z − z1 )k φ (z)

±= 0 and φ is differentiable in a neighbourhood of z1. Therefore f ² (z) = k(z − z1)k −1φ (z) + (z − z1 )k φ² (z)

so

f ² (z) f (z)

²(z) = z −k z + φφ (z) 1 This has a simple pole at z1 with residue k, because φ ² /φ is differentiable at z1 . This proves (i). Similarly in case (ii)

f (z) =

ψ (z) (z − z2)m

where ψ (z2 ) ± = 0 and φ is differentiable in a neighbourhood of z2 . Therefore f ² (z)

ψ ² (z) = (z−−mzψ)(z) + m+1 (z − z )m 2

2

262

Residues

so

f ² (z) f (z)

² (z) = z −−mz + ψψ (z) 2 which has a simple pole at z2 with residue −m. Now sum over all poles of f ² /f . C O RO L L A RY 12.13. Let γ be a simple loop in a domain D such that all points inside γ are in D. If f is differentiable in D and has no zeros on γ then the number of zeros of f inside γ is ² ² f (z) 1 dz 2π i γ f (z)

From this we deduce another important theorem: 12.14 (Rouché’s Theorem). Suppose that f and g are differentiable in a domain D that contains a simple loop γ and all points inside γ . If

THEOREM

|f (z) − g(z)| < |f (z)|

= γ (t) (t ∈ [a, b]) then f and g have the same number of zeros inside γ . Let F(z) = g(z)/ f (z). By (12.11), |1 − F(γ (t)) | < 1 (t ∈ [a, b]) (12.12)

for all z Proof.

(12.11)

Inside D, F has a zero F has a pole

⇐⇒ ⇐⇒

g has a zero f has a zero

Thus by Theorem 12.12 it is enough to prove that

²

F ²(z) dz = 0 γ F(z)

To do so, let ²(t) = F(γ (t)). By (12.12),

|1 − ²(t)| < 1

(t

∈ [a, b])

so ² lies inside the circle centre 1 radius 1, Figure 12.10. Now

²

F ² (z) dz = γ F(z)

=

²

b

a

²

²a

b

F² (γ (t)) dt F( γ (t))

² (t) dt ²(t))

= dzz ² = 2π iw(² , 0) =0 from Figure 12.10.

12.6 Exercises

Figure 12.10 Path

263

² in proof of Rouché’s Theorem.

As an example of Rouché’s Theorem in action, we give a second proof of the Fundamental Theorem of Algebra, Theorem 10.9. Let g(z) = zm

f (z) = zm

Then and for z ± = 0

|f (z) − g(z)| = |a1zm−1 + · · · + am| º º º º º1º º º |f (z) − g(z)| = º a1 ºz º zm º

The right-hand side tends to 0 as |z| then º º a1 º ºz and then

+ a1 zm−1 + · · · + am

+ ··· +

º

am ºº zm º

→ ∞, so there exists R > 0 such that if |z| > R

+ ··· +

º

am ºº 1)

dt a − cos t

What happens if a < −1? If −1 ≤ a ≤ 1? 5. Verify the following:

² π

π dt = √ (b > −1) 2 1 + b cos t b+1 ²0 ∞ π (b > −1) dx = √ (ii) 4 2 2 ²0 π 1 + x log x dx = 0 (iii) 1 + x2 ²0 ∞ x sin π x (iv) dx = π 1 − x2 −∞ ² ∞ (log x)2 π3 (v) dx = 8 −∞ 1 + x2 6. Show that (i)

² ∞

(10x)2 dx = π −∞ (x2 + 4)2 (x2 + 9)2

7. Show that

² ∞ cos 5x 0

π

dx = 3 e−5a/ x4 + a4 2a

2

´ sin

√5a + π4 2

µ

(a > 0)

12.6 Exercises

265

8. Evaluate: (i) (ii) (iii)

² 2π ²0

cos 4 t + sin4 t dt

sin3 t · cos t + cos3 t · sin t dt

²0 2π

2 cos 3 t + 3 cos 2 t dt

0

9. If n ∈ Z , n > 0, prove: (i) (ii)

² 2π

²0 2π 0

exp(cos t) cos(nt − sin t) dt

= 2π/n!

exp(cos t) sin(nt − sin t) dt = 0

10. Evaluate, by integrating suitable functions round a semicircle: ² ∞ dx (i) 2 + x4 1 + x ²0 ∞ cos mx (ii) dx (a, m > 0) 2 + a2 x 0 11. Prove that

² ∞ 0

x2 π (a > 0) dx = (x2 + a2)3 16a3

12. By integrating round a rectangle whose vertices lie at R, R letting R → ∞ , show that

² ∞ cosh(cx)

−∞ cosh(π x)

13. Prove that

² ∞ 0

dx = sec(c/2)

t a− 1(t + 1)−1 dt =

π sin π a

+ i, −R + i, −R, and

(c ∈ [−π , π ])

(a ∈ [0, 1])

by making the substitution t = ex and integrating eaz (ez + 1)−1 around the rectangle with vertices ³R, ³R + 2π i.

14. Inversion Formula for the Laplace Transform. Suppose that F is differentiable in C except for a ﬁnite number of poles, of which z1, . . . , zn satisfy re z < a and none lies on the line re z = a. If there exist M > 0, b > 0, c > 0 such that | F(z)| < M/| z| c for | z| < b, show that f (t) = lim

² a+iR

R→∞ a−iR

If F(z) = α(z2 + α 2)−1 (α

eztF(z) dz = 2π i

n ± r=1

res(ezt F(z), zr )

> 0) show that f (t) = sin αt.

15. Use real analysis to prove Jordan’s inequality: sin t t

≥ π2

(t ∈ [0, π/2])

266

Residues

Hence show that if SR (t) = Reit (t

²

lim

∈ [0, π ]) then

R→∞ SR

eimz dz = 0 z

(m > 0)

By integrating eimz/ z along the contours

²1 , Cε , ²2, SR deﬁned by: ²1(t) = t (t ∈ [−R, −ε]) Cε (t) = ei(π−t) (t ∈ [0, π ]) ²2(t) = t (t ∈ [ε , R])

prove Dirichlet’s Discontinuous Factor:

² ∞ sin mx x

0

16. Show that: z (i) z e −1

= 1 − 2z +

(ii) cosec z = (iii) cosec z = 2

1 + z

∞ ±

∞ ± n=1

dx =

⎧ ⎨ 0 ⎩

π/2 −π/2

if m = 0 if m > 0 if m < 0

2z2 z2 + 4n2π 2

∞ (−1)n2z ± n=1

z2 − n2 π 2

1 2 −∞ (z − nπ )

17. Sum the following series, where a ± ∈ Z: (i) (ii)

∞ ±

(n + a)−2

−∞ ∞ ± 0

(iii)

∞ ± 0

(iv)

(n2 + a2) −1 (2n + 1)−2

∞ ± 0

( −1)n (2n + 1)−3

18. By integrating zeibz (a2 − z2) sin π z round a suitable contour, show that

∞ ± n=1

(−1)n

n sin bn a2 − n2

sin ba = π2 sin π a (|b| < π )

12.6 Exercises

19. By considering f (z) = 1/(z − ξ ) + 1/z, show that when ξ

∞ ±

π cot πξ = 1ξ +

20.

267

±∈ Z,

ξ − n2 Using the result of Exercise 19, integrate π cot π x along a suitable contour to show that

log sin π z = log π z +

n=1

∞ ± n=1

2

log(1 − z2 /n2)

where the log is chosen to make log 1 = 0 in each term. Taking exponentials, obtain the inﬁnite product expansion of the sine function (deﬁned as the limit of suitable ﬁnite partial products, by analogy with inﬁnite sums): sin π z = π z

∞ ¿

n=1

(1 − z2/ n2 )

21. If |a| > e, use Rouché’s Theorem to prove that the equation ez = az n

has n roots with | z| < 1.

22. Find the number of zeros of the following polynomials that lie inside the unit circle: (i) z9 − 2z6 + z2 − 8z − 2 (ii) 2z5 − z3 + 3z2 − z + 8 (iii) z4 − 5z + 1

23. How many zeros of z4 + 4z3 + 6z2 − 4z + 3 lie inside the disc |z − 1| 24. Prove that however small ε

< 1?

> 0 is chosen, for all large enough n the function

1 + z−1 + (2!z2 )−1 + (3!z3 )−1 + · · · + (n!zn )−1

has all its zeros inside the disc |z|

< ε.

25. Let p(z) be a polynomial of degree n, and suppose that p(z1 ) = p(z2) = 0 with z1 ± = z2 . Show that there exists a zero of p² (z) within the circle centre 21 (z1 + z2) and radius 21 | z1 − z2| cot(π/n).

26. Residues in Reverse. Show that the residue of tanp−1 π z at z = 21 is (−1)p/2 π −1 , for integer p > 0. (Hint: integrate it round the rectangle with vertices at −iR, 1 − iR, 1 + iR, iR, then let R → ∞ and estimate sizes.) 27. Show that

² ∞ log x

dx = 0 1 + x2 ² ∞ log x π (ii) dx = − 2 2 (1 + x ) 4 0 (i)

0

13

Conformal Transformations

In mathematics we often encounter functions that preserve some structure that is of interest. For example, in Euclidean geometry rigid motions preserve lengths, angles, and areas, while changes of scale preserve the shape (but not the size) of geometric ﬁgures. Homomorphisms of groups preserve group multiplication. Arithmetic modulo n preserves addition and multiplication. Topological transformations preserve connectedness. Conversely, given an interesting class of functions, we can ask what structure they preserve. This chapter deals with a property preserved by all differentiable (equivalently analytic) complex functions, namely: angles between curves. Functions with this property are called ‘conformal’. The conformal property can be used in two directions. By studying differentiable functions, we can prove theorems about curves; by studying curves, we can prove theorems about differentiable functions. The second technique is of great importance in the advanced ‘geometric’ theory of differentiable functions, but only the ﬁrst falls within our present scope. The method has interesting applications to potential theory and ﬂuid dynamics, and we outline the beginning of these. We also consider in moderate detail several special conformal functions; in particular ‘Möbius maps’, often called ‘Möbius transformations’, which have the remarkable property of mapping circles to circles.

13.1

Measurement of Angles Since we are studying preservation of angles, we must discuss these ﬁrst, in a manner that is appropriate for this chapter. The use of a real number to measure an angle is not always entirely satisfactory, because the same angle corresponds to many different real numbers. However, if real numbers θ and φ represent the same angle, then θ − φ is an integer multiple of 2 π , and conversely, so the ambiguity is not too great. For many purposes it can be avoided by making some artiﬁcial convention, such as the requirement that −π < arg(z) ≤ π . For other purposes it is more convenient to measure angles in a natural and unambiguous way, though one that is less familiar than the real numbers.

13.1.1

π

Real Numbers Modulo 2

If x, y ∈ R we say that x and y are congruent modulo 2π , and write x≡y

(mod 2π )

13.1 Measurement of Angles

269

if there is an integer n such that x − y = 2nπ . Congruence modulo 2 π is an equivalence relation, so it partitions R into mutually disjoint equivalence classes. We denote the set of all such equivalence classes by

R/2π

For each x ∈ R let p(x) be the equivalence class (or congruence class) to which x belongs. This deﬁnes a function p:

R → R/2π

and

= {x + 2nπ : n ∈ Z} Given an angle measured by a real number θ , the same angle is measured by all θ + 2nπ (n ∈ Z), and by these real numbers only. Instead of picking one of them, we can represent the angle by the whole collection, namely p(θ ). In other words, the natural measure of an angle is not a real number, but an element of R/2π . p(x)

For instance, the real numbers

. . . , −11π/3, −5π/3, π/3, 7π/3, 13π/3, . . . all represent the same angle, and the set

{. . . , −11π/3, −5π/3, π/3, 7π/3, 13π/3, . . .} is the unique element of R/2π corresponding to this angle. 13.1.2

Geometry of R 2

Geometrically, R/2π is a circle. To see this, deﬁne q:R→C

q(x) = eix (x ∈ R)

The image of q is the unit circle in C, which we denote by

S = {z ∈ C : z = eix (x ∈ R)} = {z ∈ C : |z| = 1}

Since e2π i so

= 1, Proposition 5.3 shows that q(x) = q(y) if and only if x ≡ y (mod 2π ),

Therefore we can deﬁne

q(x) = q(y)

⇐⇒

j : R/2π

p(x) = p(y)

(13.1)

→S as follows: if r ∈ R/ 2π then r = p(x) for some x ∈ R, and we set j(r) = q(x) By (13.1) this is independent of the choice of x ∈ R, and j is a bijection. Thus the elements of R/2π are in natural one-to-one correspondence with the points of a circle.

270

Conformal Transformations

This bijection preserves continuity in a natural manner. If X ⊆ C we say that a function f : X → R/2π is continuous if and only if the composite function j ◦ f : X → S is continuous in the usual sense, considering S as a subset of C. This accords with the intuitive idea of geometric continuity if we think of R/2π as a circle. (In topological terms, we use j to transfer the usual topology of S to R/2π , deﬁning U ⊆ R/ 2π to be open if and only if j(U) is open in S . Continuity of maps f : X → R/2π in this topology is equivalent to what we have just deﬁned.)

13.1.3

Operations on Angles We can deﬁne addition and subtraction of elements of R/ 2π , corresponding to geometric addition and subtraction of angles. Let r, s ∈ R/2π . Pick x, y ∈ R such that p(x) = r, p(y) = s, and deﬁne r + s = p(x + y) r − s = p(x − y) As usual we can verify that these deﬁnitions do not depend on the choice of x and y, using (12.1). We can restate the above in group-theoretic terms. The set G = {2nπ : n ∈ Z} is a subgroup of the additive group of real numbers, hence a normal subgroup since the latter is abelian. The quotient group R/G is what we have called R/2π . As an analogy, consider the deﬁnition of the integers Zn modulo n as the quotient group Z /H where H = {kn : k ∈ Z}. In Zn we can also deﬁne multiplication. You should verify that this does not work for R/2π . It fails because 2π is not an integer. However, we can sensibly multiply an angle in R/2π by an integer, since this reduces to repeated additions or subtractions.

13.1.4

π

The Argument Modulo 2

For z ∈ C \ { 0} we can now deﬁne a ‘mod 2π ’ version of arg z, namely arc z

= p(arg z) ∈ R/2π

The notation is deliberately chosen to resemble ‘arg’, and also reminds us that angles are involved. The advantages of ‘arc’ over ‘arg’ are twofold. First, there is no ambiguity. Second, and more important, the function arc :

C \ {0} → R/2π

is continuous. This is false for arg, for as z moves across the negative real axis, arg z jumps from near π to near −π , because of the convention that −π < arg z ≤ π . In contrast, since p(−π ) = p(π ), this defect is not shared by arc. In fact, if z = reiθ , r > 0, then j(arc(z)) = eiθ . Using this, or directly, it is not hard to verify that arc(z1z2 ) = arc z1 + arc z2

(13.2)

for all z1, z2 ∈ C \{0}. Again if we use arg this is true only up to integer multiples of 2π .

13.2 Conformal Transformations

271

Thus we have two distinct ways to represent an angle: as a uniquely deﬁned element of R/2π (an equivalence class of reals modulo 2 π ), or as a real number chosen from this class, unique only up to integer multiples of 2π . Both are useful, and there are advantages in passing between them at will. We do this in the proof of our next result. 13.1. If γ : [a, b] → C is a path and γ ± (t0) exists and is non-zero for some [a, b], then γ has a tangent at z0 = γ (t0) making an angle arc γ ± (t 0) with the real

LEMMA

t0 ∈ axis.

Proof. Let γ (t) = x(t) + iy(t). Then the required angle θ belongs to the congruence class modulo 2π of tan−1 (y± (t 0)/x± (t0 )) = arg(x± (t 0) + iy± (t0))

= arg γ ± (t0)

Taking congruence classes, we get

θ = arc γ ± (t 0) In future we will be less explicit about passing between

R and R/2π .

D E FI N I T I O N 13.2. If γ1 and γ2 are two paths meeting at z0 = γ1(t 1) = derivatives γ1± (t1), γ2± (t2 ) ² = 0, then the angle between γ1 and γ2 at z0 is

γ2(t 2) having

θ = arc γ1± (t1 ) − arc γ2± (t2) ∈ R/2π

as in Figure 13.1.

Figure 13.1 Angle between two intersecting curves.

13.2

Conformal Transformations In this section we consider functions f : D → C, where D is a domain. It is convenient to distinguish the two copies of C by using (x, y) as coordinates in D and (u, v) as coordinates in the image C. As usual we let z = x + iy, and we set w = u + iv. (The classical notation for complex functions was w = f (z) before set theory came along.) If f is differentiable on D we have f (x + iy) = u(x, y) + iv(x, y)

272

Conformal Transformations

where u and v are real-valued functions of two real variables x, y. Hence f deﬁnes a function from the subset D of the (x, y)-plane to the (u, v)-plane. A smooth path γ in D, with

γ (t) = x(t) + iy(t)

(t ∈ [a, b])

is mapped by f to a path f γ (t) = f ( γ (t))

= u(x(t), y(t)) + iv(x(t), y(t))

(t

∈ [a, b])

in the (u, v)-plane.

13.3. Until now we have written composition as f ◦ γ , but from now on we omit the circle in formulas when there is no confusion with the product. This avoids undue proliferation of circles.

REMARK

Suppose that z0

= γ (t 0) and γ ± (t0 ) ²= 0 for some t0 ∈ [a, b]. By the chain rule, ( f γ )± (t0) = f ± (γ (t 0))γ ± (t0 ) = f ± (z0)γ ± (t 0)

Therefore arc (( f γ )± (t0 )) = arc ( f ± (z0 )γ ± (t0 ))

= arc f ± (z0) + arc γ ± (t0 )

(13.3)

by (13.2). Suppose that γ1 and γ2 are two paths through z0, say z0 = γ1(t 1) = γ2 (t2 ). Then (13.3) implies that f γ1 and f γ2 meet at the same angle as γ1 and γ2 . Geometrically this is clear since both tangents are turned through the same angle arc f ± (z0 ). Alternatively, compute: arc (( f γ1)± (t1 )) − arc (( f γ2)± (t2 )) = arc f ± (z0) + arc γ1± (t1) − arc f ±(z0 ) − arc γ2± (t2 )

= arc γ1± (t1 ) − arc γ2± (t2) : D → C that preserves angles between paths at a

13.4. A function f point z0 is conformal at z 0. If f is conformal at all z0 ∈ D we say it is conformal.

D E FI N I T I O N

The terms conformal function, conformal map, conformal mapping, and conformal transformation all mean the same thing. The fourth is traditional, the third is going out of fashion, the second is convenient, and the ﬁrst agrees with much current terminology. One advantage of the fourth is that it focuses attention on how paths and other geometric ﬁgures transform under f . Observe that we have proved: THEOREM

13.5. Let f : D

such that f ±(z0 ) ² = 0.

→ C be differentiable. The f is conformal at all z0 ∈ D

If f ±(z0 ) = 0 then f may not be conformal at z0 . For example, if f (z) = z2 then the positive half of the real axis and the ‘positive’ half of the imaginary axis ({iy : y > 0}) meet at right angles. They transform into the positive half of the real axis and the negative half of the real axis, with an angle of π .

13.2 Conformal Transformations

273

In fact, if z0 is a zero of f ± of order m then the angle between paths meeting at z0 is multiplied by m + 1 on transforming by f . We can ﬁnd out a little about how f affects lengths. If z0, z ∈ C and f is differentiable at z0, then the ratio of the distances between f (z) and f (z0) and between z and z0 is

|f (z) − f (z0 )| = ±±± f (z) − f (z0) ±±± ± z − z0 ± |z − z0| which tends to f ±(z0 ) as z → z0. So near z0 distances are approximately multiplied by |f ± (z0)|. Some special cases illustrate the previous analysis.

Example 13.6. f (z) Here

= z3 . u(x, y) + iv(x, y) = (x + iy)3

= (x3 − 3xy2) + i(3x2y − y3 )

so that u(x, y) = x3 − 3xy2 v(x, y) = 3x2 y − y3

Consider the paths

These are respectively the lines x

γ1(t) = 1 + it γ2(t) = t + i = 1, y = 1, which meet at right angles. The paths

±1 = f γ1 and ±2 = f γ2 are given by ±1 (t) = (1 − 3t2 ) + i(3t − t 3) ±2 (t) = (t3 − 3t) + i(3t 2 − 1) These curves are sketched in Figure 13.2. As expected, ±1 and ±2 meet at right angles at (−2, 2). Example 13.7. f (z) This time

= 1/z. u(x, y) =

If c > 0, the circle

x

+ y2 −y v(x, y) = 2 x + y2

γc (t) = ceit

x2

(t

∈ [0, 2π ])

(13.4)

274

Conformal Transformations

Figure 13.2 Transformed paths for Example 13.6 also meet at right angles.

Figure 13.3 Images of circles centred at the origin for Example 13.7.

transforms into

±c (t) = c−1 e−it

(t

∈ [0, 2π ])

(13.5)

Hence the system of concentric circles (13.4), as c varies, maps into the system of concentric circles (13.5), Figure 13.3. Further, the lines y = kx (k ∈ R) through the origin are given by

δk (t) = t + kit and transform to

²k (t) = (t + kit)−1 = 1 +1 k2 1t − 1 +ikk2 1t

which also represents lines through the origin. Now γc and and so do their transforms ±c and ²k , Figure 13.4. Example 13.8. f (z) Here

= sin z. u(x, y) = sin x cosh y

v(x, y) = cos x sinh y

δk all meet at right angles,

13.2 Conformal Transformations

275

Figure 13.4 Paths for Example 13.7 and their transforms all meet at right angles.

Figure 13.5 Example 13.8: Cartesian coordinate grid transforms into a system of confocal

hyperbolas and ellipses. Again all curves meet at right angles.

The lines x = c (c ∈ R) transform into confocal hyperbolas u2 sin2 c

2

− cosv 2 c = 1

and the lines y = d (d ∈ R) transform into confocal ellipses u2 cosh2 d

+

v2 sinh 2 d

=1

as in Figure 13.5. The two systems of straight lines form a grid in Cartesian coordinates, crossing at right angles. The hyperbolas and ellipses also meet at right angles, except at points where f ± (z) = 0, namely f (z) = ³1. These are the common foci of the ellipses and hyperbolas.

276

Conformal Transformations

13.3

Critical Points Theorem 13.5 proves that a differentiable complex function is conformal at any point z0 such that f ± (z0) ² = 0. We now consider what happens when f ± (z0 ) = 0. D E FI N I T I O N

13.9. Let f be a differentiable complex function. A point z0 is a regular

point if f ± (z0) ² = 0. It is a critical point or singular point if f ± (z0 ) = 0.

Even though a complex function f need not be conformal near a critical point, the local geometry still has considerable structure. To unravel what happens at a critical point, consider the local Taylor expansion: f (z0 + h) = f (z0 ) + hf ± (z0 ) + · · · + hn f (n) (z0)/n! + · · · This consists of two parts. The constant part is w0 = f (z0 ), which calculates where the point z0 in the z-plane is mapped in the w-plane. The variable part f (z0 + h) − f (z0 ) = hf ± (z0 ) + · · · + hn f (n) (z0)/n! + · · ·

speciﬁes how the function behaves near w0 = f (z0 ). In particular, the behaviour is controlled by the ﬁrst non-zero derivative f (n (z0). This motivates: 13.10. A differentiable function f : D → C is of order n at z0 (where n ≥ 1) if f (n) (z0 ) ² = 0, and f (k) (z0) = 0 for 1 ≤ k < n.

D E FI N I T I O N

D,

A function of order 1 satisﬁes f ± (z0) ² = 0, and in general, the Taylor series at a ﬁxed point z0 for a function of order n has the form f (z0 + h) = f (z0) + hnf (n) (z0) /n! + · · ·

(13.6)

Examples 13.11. (i) f (z) zn for integer n 1 is of order n. (ii) sin z z z3 3 is of order 1. (iii) cos z 1 z2 2 z4 4 is of order 2.

= ≥ = − / ! + ··· − = / ! − / ! + ···

Equation (13.6) indicates that the behaviour of a function near a critical point is given by the order of the corresponding zero of the derivative. Indeed, a function of order n ≥ 1 can be written as: f (z0 + h)

= f (z0 ) + khn + higher order terms in h, where k = f (n) (z0)/n! ² = 0

This gives an approximation of practical value in applications: f (z0 + h) − f (z0 ) ≈ kh n for suitably small h

(13.7)

The size of h required to give a decent approximation depends on the context. As z moves from z0 − h to z0 + h, the difference f (z) − f (z0 ) in (13.7) changes approximately from k(−h) n to kh n. The behaviour depends on whether n is even or odd. For n odd,

13.3 Critical Points

277

it moves from −khn to kh n, passing through zero, and for n even it travels from kh n to zero, then back again to khn . A more precise sense of the behaviour can be obtained by writing h in polar coordinates, h = reiθ . For suitably small h, (13.7) gives f (z0 + reiθ ) − f (z0 ) = krneinθ

+ ···

= |f (n) (z0)|/n!)

(k

so

|f (z0 + rei θ ) − f (z0)| ≈ krn and arc ( f (z0 + reiθ ) − f (z0))

≈ nθ + α

where α = arc ( f (n) (z0 ))

For small values of h, f transforms a tiny portion of the z-plane near z0 to the w-plane near w0 = f (z0 ), scaling r to krn and rotating the original angle θ to nθ +α, Figure 13.6. Rotating the line from z0 to z0 + h in the z-plane through a further angle φ rotates the corresponding line in the w-plane to n(θ + φ ) + α , so the angle between the transformed lines in the w-plane is nφ. The sector of the circle radius r, angle φ in the z-plane is scaled to a sector in the w-plane, radius krn , with angle expanded to nφ , Figure 13.7. For n = 1 the point is regular, and f is conformal. For n > 1, the function f transforms z0 to w0 = f (z0), and a point distance r from z0 is scaled approximately to krn from w0 in the w-plane, while the angle between two paths through z0 changes from φ in the z-plane to nφ in the w-plane.

z0 +h r

z0

θ

z-plane

f for suitably small h

scale to kr n

f(z0 +h) rotate to nθ + α

w0 w-plane

Figure 13.6 The tangent map translating, rotating and scaling from z 0 to w0.

Figure 13.7 Transforming a sector of a circle centre z0, radius r.

278

Conformal Transformations

13.4

Möbius Maps For ﬁxed a, b, c, d ∈ C, the function f (z) =

az + b cz + d

²= 0)

is called a Möbius map or bilinear map. (Again, ‘map’ can be replaced by ‘function’, ‘mapping’, or ‘transformation’.) These maps have a number of important properties, and we discuss some of them. First, note that f is differentiable for z ² = −d/c, and f ± (z) =

ad − bc (cz + d)2

²= 0

since ad − bc ² = 0, again provided z ² = −d/c. Hence f is conformal throughout the domain C \ {−d/c} on which it is deﬁned. Now suppose that Az + B f (z) = (AD − BC ² = 0) Cz + D is a second Möbius map. Then the composite gf (z) =

(Aa + Bc)z + (Ab + Bd) (Ca + Dc)z + (Cb + Dd)

is another Möbius map, because (Aa + Bc)(Cb + Dd) − (Ab + Bd)(Ca + Dc)

So composing two Möbius maps always gives a Möbius map.

13.4.1

Möbius Maps Preserve Circles A remarkable, and useful, property of such maps is that they transform circles (or straight lines) into circles (or straight lines). To save breath, let us agree that in this section a ‘circle’ may be either a circle or a straight line. (We can think of a straight line as a circle of inﬁnite radius, remembering that ∞ is a fairly respectable concept in complex analysis.) Suppose p, q ∈ C with p ² = q, and let k > 0. The equation

|z − p| |z − q| = k

(13.8)

is satisﬁed by those points whose distances from p and q are in the ratio k. It is well known in Euclidean geometry, and can easily be checked using coordinates, that if k ² = 1 such points lie on a circle, and if k = 1 they lie on a straight line. Here is a quick proof. By scaling and a rigid motion, we can choose coordinates in the plane so that p = (0, 0) and q = (1, 0). If (x, y) is distance kd from p, and distance d from q, then

²

x2 + y2

=k

²

(x − 1)2 + y2

13.4 Möbius Maps

279

Hence x2 + y2 = k2(x2 + y2 − 2x + 1) which implies

³

x−

k2

´2

k2 − 1

k2

+ y + k2 − 1 − 2

³

k2

´2

k2 − 1

=0

which is the equation of a circle. If we put w

az + b = f (z) = cz +d

it is easy to verify that z= Now (13.8) takes the form

−dw + b cw − a

± az+b ± ± cz+d ± az+b ± cz+d

which simpliﬁes to

(ad − bc ² = 0)

±

− p ±± ±=k − q±

|w − P| |w − Q| = K

(13.9)

where

= (b + pa)/ (d + pc) Q = (b + qa)/ (d + qc) K = k|d + qc|/|d + pc| P

Hence (13.9) also represents a circle. This proves that the Möbius map f transforms circles into circles.

13.4.2

Classiﬁcation of Möbius Maps The above calculation, while it veriﬁes the assertion, is not as instructive as we might hope, and the following approach has its advantages. We consider several special types of Möbius map that have especially simple forms, which can easily be seen to transform circles into circles. Then we show that a general Möbius map can be obtained by composing these types. It is perhaps not so surprising that this approach works, in view of the composition property of Möbius maps. The decomposition can be seen as analogous to the wellknown fact that any rigid motion of the Euclidean plane can be obtained by composing a translation, a rotation, and a reﬂection. We begin with the special types.

280

Conformal Transformations

Translation: w = z + k (k ∈ C). This corresponds geometrically to moving points re k to the right and im k upwards, and clearly preserves the shape of geometric ﬁgures, in particular circles. Rotation: w = eiθ z (θ ∈ R). All points rotate round the origin through angle θ . Again this preserves the shape of geometric ﬁgures. Magniﬁcation: w = hz (h > 0). This produces a change of scale. If h < 1 it shrinks rather than magniﬁes, but such pedantry is irrelevant and we still use the term ‘magniﬁcation’. It therefore maps geometric ﬁgures to similar geometric ﬁgures. Inversion: w = 1/z. Devotees of ‘inversive geometry’, now largely out of fashion, will recognise the corresponding ‘geometric inversion’, which is well known to preserve circles. However, circles can map to straight lines, or straight lines to circles. The rest of us can check this by a coordinate calculation similar to the one above. We can now state: 13.12. Every Möbius map can be obtained by composing a translation, an inversion, a magniﬁcation, a rotation, and another translation.

THEOREM

∈ C and ad − bc ²= 0, where c ²= 0. Deﬁne t1(z) = z + d/c (translation) j(z) = 1/z (inversion) ± ± ± ad − bc ± ± z (magniﬁcation) ± m(z) = ± c2 ± (bc − ad)| c| 2 r(z) = |ad − bc|c2 z (rotation) t2(z) = z + a/c (translation)

Proof. Suppose that a, b, c, d

It is routine to verify that (t2rmjt 1)(z) If c = 0 deﬁne

az + b = cz +d

t1 (z) = z + b/a (translation ±a± ± ± m(z) = ± ± z (magniﬁcation) d± ± a ±d± r(z) = ±± ±± z (rotation) d a

²= 0 since c = 0, hence a ²= 0, d ²= 0. Now az + b (rmt1 )(z) = cz + d

C O RO L L A RY

13.13. Every Möbius map transforms circles into circles.

13.5 Potential Theory

281

There are other functions f : C → C that preserve circularity, the most obvious being complex conjugation (which is not differentiable). A theorem of Constantin Carathéodory asserts that every such function is either a Möbius map, or a Möbius map composed with conjugation. No differentiability assumptions are needed for this theorem. We do not prove it here.

13.4.3

Extension of Möbius Maps to the Riemann Sphere There is a natural way to extend Möbius maps to the Riemann sphere by examining their behaviour as z → ∞ . P RO P O S I T I O N 13.14. Any Möbius map can be extended uniquely to the Riemann sphere so that it is differentiable at every point.

Proof. By continuity we can ﬁnd this extension by letting z verify differentiability. If f (z) = then letting z → ∞ we must deﬁne

az + b cz + d

f (∞)

=

µ a

Differentiability at ∞ is an easy exercise.

c

→ ∞, but we must then

²= 0)

if c ² = 0 if c = 0

When extended in this manner, f is a bijection from C ∪ {∞} to itself. If we identify C ∪{∞} with the Riemann sphere S2 , then circles in C remain circles on S2 , and straight lines in C also become circles on S 2. It is instructive to work out the effect of the basic types of Möbius map on S2 . We omit the details.

13.5

Potential Theory We now turn – brieﬂy – to classical applications of complex analysis. These were instrumental in convincing mathematicians that the subject was worth taking seriously, before the basic concepts had been made rigorous. We discuss potential theory, a general area of mathematical physics with applications to gravitation electrostatics, magnetism, and ﬂuid ﬂow.

13.5.1

Laplace’s Equation The two-dimensional Laplace equation

∂ 2φ + ∂ 2 φ = 0 ∂ x2 ∂ y2

282

Conformal Transformations

for a function φ(x, y) is important in potential theory, with applications in particular to ﬂuid dynamics. It is closely connected with complex function theory, as we now demonstrate. Let f : D → C be differentiable, with z = x + iy, and write f (z) as usual. Then f ± (z) =

= u(x, y) + iv(x, y) ∂u ∂v ∂v ∂u ∂ x + i ∂x = ∂y − i ∂y

as in Section 4.2. By Theorem 10.3 f ±± exists throughout D. If we let f ± (z) the Cauchy–Riemann Equations of Theorem 4.12 state that

∂ U = ∂V ∂x ∂y Therefore U so we get

= ∂∂ xu = ∂∂ yv

= U + iV then

∂V = −∂ U ∂x ∂y = ∂∂ vx = − ∂∂uy

V

∂ 2 u = ∂ ³ ∂ u ´ = ∂ U = ∂ V = − ∂ ³ ∂ u ´ = − ∂ 2φ ∂ x2 ∂ x ∂ x ∂x ∂ y ∂y ∂y ∂ y2

Therefore u(x, y) satisﬁes the Laplace equation. Similarly, so does v(x, y). For instance, consider f (z) = zez . Then u(x, y) = xex cos y − yex sin y

v(x, y) = yex cos y + xex sin y

and it may be veriﬁed directly that these functions satisfy Laplace’s Equation. Solutions of Laplace’s equation are called harmonic or potential functions. Pairs of functions u, v obtained from a differentiable function by the above method are called harmonic conjugates. The lines u = constant, v = constant, are orthogonal (mutually perpendicular) in the (u, v)-plane, so by conformality the curves u(x, y) = constant

v(x, y) = constant

are orthogonal in the (x, y)-plane. In potential theory, if u is harmonic, the lines u(x, y) = constant are called equipotential lines, and the set of orthogonal curves v(x, y) = constant are called streamlines. In the case when Laplace’s equation describes ﬂuid ﬂow, streamlines are the paths along which the ﬂuid ﬂows. If we are given u in a domain D and wish to ﬁnd the streamlines given by v, we can often use complex integration. For a ﬁxed point z0 ∈ D, and any z1 ∈ D, we have f (z1 ) =

z1

z0

f ± (z)dz =

z1

z0

³

∂ u − i ∂ u ´ dz ∂x ∂ y

13.5 Potential Theory

For example, if u(x, y) convenience we have

=

f (z1 ) =

x2

¶ 0

− y2 , which is harmonic, then picking z0 = z1

(2x + 2iy)dz =

z1

0

283

0 for

2zdz = z21

Hence f (x + iy) = (x + iy)2 = (x2 − y2 ) + i(2xy). So the streamlines have equations 2xy = constant, or equivalently xy = constant

Often, as in this case, we can guess what f , hence v(x, y), ought to be – a process digniﬁed by the term ‘inspection’.

13.5.2

Design of Aerofoils Conformal maps and complex function theory played a part in the design of aircraft in the early days of aviation (and, in a more sophisticated way, still do, although most modelling is now done using the numerical methods of computational ﬂuid dynamics). In particular, the transformation from the z-plane to the w-plane given by w−2 w+2

=

³

z−1 z+1

´2

maps a circle in the z-plane, passing through −1 and containing +1 in its interior, into a ‘bent teardrop’ shape as in Figure 13.8 that resembles a cross-section of an aircraft’s wing. This is known as a Joukowski aerofoil, deﬁned by a Joukowski transformation, after Nikolai Zhukovsky, who discovered the transformation.

Figure 13.8 Left: Flow past circle (with appropriate circulation added). Right: Circle transformed

to Joukowski aerofoil, with transformed ﬂow.

It is used as follows. It is quite easy to solve the Laplace equation and ﬁnd streamlines for the ﬂow of a ﬂuid round a circular disc. Now apply the Joukowski transformation: the disc maps to an aerofoil, and the streamlines round the disc map to the streamlines round the aerofoil. In order to make the ﬂow behave in a sensible physical manner at the sharp edge of the aerofoil, the ﬂow past the circle is modiﬁed by adding a ‘circulation’ term that ﬂows round the circle. From this we can calculate properties of the ﬂow; in particular, the amount of ‘lift’ imparted to the aircraft. More subtle transformations, such as the Karmann–Trefftz transform

284

Conformal Transformations

w−n w+n

=

³

z−1 z+1

´n

give more accurate information.

13.6

Exercises 1. Sketch the image of the set D = {z ∈ C : re z > 0, im z > 0, |z| < 1} under the conformal map f (z) = 1/z. Draw the images of the lines in D parallel to the coordinate axes. 2. Show that the map

· ·π 1 + k2 exp −i b 2

¸ ¸

+ tan−1 k z transforms the strip between the lines y = kx, y = kx + b into that between x = 0 and x = 1. Find a conformal map f of the annulus 2 < | z| < 5 on to the annulus 4 < | z| < 10, such that f (−5) = 10. (These points lie on the boundary so f must also be deﬁned there.) Find another with f (5) = 4. f (z) =

3.

4. With a suitable choice of the square root, show that f (z) = maps the exterior of the parabola y2 x > 0, Figure 13.9.

√z − p − i√p

= 4px (p > 0) onto the right-hand half-plane

5. Find a conformal map that sends the interior of the right-hand branch of the hyperbola x2 − y2 =

λ2

to the upper half-plane y > 0, Figure 13.10. (Hint: what is re z2 ?)

6. Show that the semicircle | z| < 1, re z

> 0 is mapped by f (z) = z2 + z

Figure 13.9 Conformal map of interior of parabola to half-plane.

13.6 Exercises

285

Figure 13.10 Conformal map of interior of hyperbola to half-plane.

Figure 13.11 Conformal map for Exercise 6.

Figure 13.12 Region for Exercise 7.

to the region bounded by the parabola x polar coordinates is

= −y2 and the curve whose equation in

r = 2 cos( θ/3) ( |θ| ≤ 3π/4) See Figure 13.11.

7. Show that for suitably deﬁned square roots, when a, b, c > 0,

√2 2 √ 2 2 z + c + a +c √ f (z) = √ bz 2 + c2 − z2 + c2 maps the plane with the segments between −a, and b, and −ic and ic removed, to ¹

the upper half-plane. See Figure 13.12.

286

Conformal Transformations

–1 + i

i

–1

1+i

1 1

–1 – i

–i

1–i

Figure 13.13 Left: Region for Exercise 8. Right: Exterior of unit circle.

Figure 13.14 Region for Exercise 9.

8. Show that for suitably deﬁned square roots, f (z) = º

1

2+

³²º

5

z4

+4+ 2+

²º

z4

+4−

´

5

maps Figure 13.13 (left) to the exterior of the unit circle, Figure 13.13 (right). 9. Find a conformal map from Figure 13.14 to the upper half-plane. (Hint: compare Exercise 7.) 10. Show that f (z) =

»

cos π z − cosh π h 1 + cos π z

maps Figure 13.15 to the upper half-plane. 11. Find a conformal map from Figure 13.16 to the upper half-plane. 12. Find the Möbius map that sends −1, ∞ , i respectively to: (i) i, 1, 1 + i (ii) ∞ , i, 1 (iii) 0, ∞ , 1

13. Find the general form of a Möbius map that: (i) Transforms the upper half-plane to itself. (ii) Transforms the upper half-plane to the lower half-plane.

13.6 Exercises

287

Figure 13.15 Region for Exercise 10.

Figure 13.16 Region for Exercise 11.

(iii) (iv) (v) (vi)

Transforms the upper half-plane to the right half-plane. Preserves the unit circle. Preserves the coordinate axes. Transforms the upper half-plane to the interior of the unit circle.

14. Invariance of Cross Ratio. The cross ratio of four complex numbers z1, z2 , z3 , z4 is (z1 − z2)(z3 − z4 ) (z1 − z4)(z3 − z2 )

az + b be a Möbius map with ad cz + d j = 1, 2, 3, 4. Prove that Let

µ(z) =

(w 1 − w2)(w3 − w4 ) (w 1 − w4)(w3 − w2 )

− bc ²= 0, and let wj = µ(zj ) for

)(z3 − z4) = (z(z1 −− zz2)(z −z ) 1

4

3

(Hint: ﬁrst work out how w1 − w2 relates to z1 − z2 .)

2

15. Let f be a Möbius map. A ﬁxed point of f is a point z such that f (z) = z. Prove that f has at most two distinct ﬁxed points, including ∞ . If f has a unique ﬁxed point (including ∞) it is said to be parabolic. Prove it can then be written in the form 1 = +h f (z) − z0 z − z0 1

(z0 ² = ∞)

288

Conformal Transformations

or

f (z) = z + h

(a translation)

If f has two distinct ﬁxed points z1 , z2 , show that it can be written in the form

or

f (z) − z1 f (z) − z2

= k zz −− zz1 2

(z1, z2

²= ∞)

f (z) − z1

= k(z − z1) (z2 = ∞) Such a map is said to be hyperbolic if k > 0, elliptic if k = loxodromic if k = aeiα , where a ² = 1 is real and α ² = 0.

ei α (α

²=

0), and

16. With the deﬁnitions of Exercise 14, prove the following: (i) Any Möbius map (az + b)/(cz + d) is equal to one for which ad − bc = 1. (ii) Having ensured this, if a + d is real then the map is elliptic if |a + d| < 2, hyperbolic if |a + d| > 2, and loxodromic if |a + d| = 2. (iii) If a + d is not real, the transformation is loxodromic. 17. Show that

u(x, y) = x3 − 3xy2

→ R such that f (x + iy) = u(x, y) + iv(x, y)

is harmonic. Find a harmonic function v : R2

is a differentiable complex function. Prove that v is unique up to the addition of a real constant.

18. For f (z) = 1/z, write f (z) = u(x, y) + iv(x, y). Sketch the level curves u(x, y) = constant, v(x, y) = constant. If a level curve of u meets a level curve of v, what is the angle between them? Is there an easy way to see this, and if so, what is it? 19. Find the most general cubic polynomial u(x, y) = ax 3 + bx2 y + cxy2 + dy3

(a, b, c, d ∈ R)

that is harmonic. Find a differentiable function with u as its real part. 20. Verify that the Joukowski transformation does, as claims above, give rise to an aerofoil shape. Look up pages 131–134 of Kyrala [12] and see how to compute streamlines round it. A more modern book is Anderson [1]. The NASA applet at www.grc.nasa.gov/WWW/K-12/airplane/map.html is also worth investigating.

14

Analytic Continuation

When Weierstrass started his programme to rigorise complex analysis he based it on power series. Because these have well-behaved convergence properties, and can be differentiated or integrated term by term, they provide a tool of great technical value. However, this tool has limitations, which we illustrate in the ﬁnal section of this chapter: many important functions cannot be represented by a single power series – mainly because power series converge on discs. This limitation can be overcome by the method of ‘analytic continuation’, which lets us extend the domain of a complex function, subject to suitable conditions. It turns out that such an extension is not always unique, and the problem of describing the different possibilities and how they are related leads to a remarkable geometric concept, known as a ‘Riemann surface’ after its inventor. In this chapter we discuss these topics and related ones. Here we use the term ‘analytic’ rather than ‘differentiable’, to emphasise the power series viewpoint – but you should remember that in the complex case these terms are synonymous.

14.1

The Limitations of Power Series We illustrate the problem for the function f (z) =

1 1 − z2

It would be possible to use a simpler example, such as 1/ z, but it is more appropriate to work with something a little less special, where the general problem is more sharply deﬁned This function is analytic in C \ {−1, 1 }, and has simple poles at −1 and 1. Its power series expansion about z0 = 0 is 1 + z2 + z4 + z6 + · · ·

(14.1)

which converges for |z| < 1 but diverges for |z| > 1. Thus if we require functions to be deﬁned on domains, the series (14.1) at best represents f on an open disc D1

= { z ∈ C : | z| < 1 }

Thus (14.1) tells us about a small part of f (directly, at least, although it has useful indirect implications, as we shall see).

290

Analytic Continuation

We can begin to get round this problem by choosing a different z0, for example z0 = i. To ﬁnd the Taylor series for f (z) around z0 = i it helps to rewrite f in the form f (z) = Let w

1 2

±

= z − i, so that z = w + i. Then f (z) =

1 2

=

1 2

= =

± ³

1 1+w+i

+

±

1

1+i

1+

∞ 1¶ 1 ± µ n=0

2 1+i

+ 1 1+z 1− z

1 1− w−i w

1+i

µ 1 (−1)n 2(1 + i) n=0

1

±

² −1 w

1+i

²n

D2

² ±

1

+ 1−i

²n

1+

1

+ 2(1 − i)

−1 + 1 1+i 1−i √

The series (14.2) has radius of convergence to the nearest pole of f , so it converges on

²

±

1

1−i

w

1−i

² −1´

∞ ± w ²n µ 1−i

n=0

²n·

(z − i)n

(14.2)

2, because that is the distance from z0

= {z ∈ C : | z − i |
1|). For a more complicated function with more poles (or other singularities) we may need more discs in a chain, because the discs have to ‘push between’ the singularities, see Figure 14.3. In a sense this is the reason for the limitation of power series: discs are too blunt an instrument to penetrate beyond the singularities into more distant territory.

14.3 Analytic Continuation

293

When we formalise these ideas, as in the next section, it becomes clear that the restriction to power series is inessential. This is often the way in mathematics: the solution to a special problem turns out to apply in a much more general setting. The special problem plays the useful role of a psychological springboard, hurling our thoughts into higher realms.

14.3

Analytic Continuation The simplest type of analytic continuation is direct analytic continuation, extending a function from one domain to an overlapping domain. Repeating this process we get indirect analytic continuation. We consider these in turn.

14.3.1

Direct Analytic Continuation D E FI N I T I O N 14.2. If f1 is analytic on a domain D1 and f2 is analytic on a domain D2 , where D1 ∩ D2 ± = ∅ and f1(z) = f2(z) for all z ∈ D1 ∩ D2, then f2 is a direct analytic continuation of f 1 to the domain D2 , see Figure 14.4.

Such an f2 must be unique by Lemma 14.1. As a simpler example than that in the previous section, take f1 (z) =

∞ µ n=0

zn

(| z|

f2 (z) = 1/(1 − z)

< 1) (z ∈ C \ {1})

Then f2 is a direct analytic continuation of f 1. Whereas f1 is deﬁned only in the interior of the unit disc, f2 is deﬁned on the whole of C except at 1. Before going on to the general case, a series of overlapping domains, it is worth dealing with another phenomenon. Sometimes D1 may be such that f1 has no direct continuation to any D2 not contained in D1 . In this case the boundary of D1 is called a natural boundary for f 1. Now the limitation does not stem from our choice of tools, but from intrinsic properties of f1 : it so happens that D1 is the end of the line as far as analytic continuation of f1 is concerned. The standard example of this phenomenon is

Figure 14.4 Direct analytic continuation.

294

Analytic Continuation

f (z) =

∞ µ zn !

(14.5)

n=0

which converges (so is analytic) for | z| < 1. We prove: 14.3. The unit circle S the function f deﬁned by (14.5).

P RO P O S I T I O N

= {z ∈ C : |z| = 1} is a natural boundary for

Proof. Let z0 = eip/ q for p, q, ∈ Z, q ≥ 1. Then zn0! = 1 for all n ≥ q. We show ﬁrst that as z → z0 then f (z) → ∞. Let z = rz0 where 0 < r < 1. Then f (z)

=

∞ µ n=0

(rz0 )n!

= (1 + rz0 + · · · + r (q−1)! z0(q−1)! ) + = g(r) + h(r), say.

Note that h(r) has the above form because zn0! N ≥ 0,

n=q

= 1 for all n ≥ q. For any ﬁxed integer

q+N

µ n! r → N + 1 as r

So for some ε

∞ µ rn !

n=q

→1

> 0, if 1 − ε < r < 1, then q+N

µ n! 1 r ≥ 2N n=q

Then

q+N ∞ µ µ n! 1 n! h(r) = r ≥ r ≥ 2N

so that h(r) → ∞ as r

n=q

n=q

→ 1, and h(r) → ∞ as z → z0. On the other hand, (q−1)! g(z) = 1 + z0 + · · · + + r(q−1)! z0 as z → z0

Therefore

lim f (z) = ∞

z →z 0

(14.6)

Now, suppose that F is a direct analytic continuation of f to a domain D2 not contained in D1 . Then ∂ D1 ∩ D2 is open in ∂ D1 = S, so it contains some point z0 = eip/q for p, q, ∈ Z, q ≥ 1 since such points are clearly dense on the unit circle. There is a small disc Dε round z0 such that Dε ⊆ D2 . Now for 1 > r > 1 − ε we have rz0 ∈ Dε , so that F(rz0) As r → 1, by (14.6). But F is analytic in D2 , so

= f (rz0 )

F(rz0 ) → ∞

14.3 Analytic Continuation

295

Figure 14.5 Sequence of points tending to z0 .

Figure 14.6 Sequence of overlapping domains.

F(rz0)

→ F(z0 )

as r → 1, by continuity. This is a contradiction; therefore no direct analytic continuation of f to D2, not contained in D1, is possible. Figure 14.5 illustrates the geometry of this proof. Informally, the point is that the singularities e ip/q are so close together that there is no room to squeeze a disc between them, onto which f might be analytically continued.

14.3.2

Indirect Analytic Continuation We now return to the question of performing a sequence of direct analytic continuations, one after the other. D E FI N I T I O N 14.4.

Let D1, . . . , Dn be domains such that Dr ∩ Dr +1

±= ∅

(r

= 1, . . . , n − 1)

as in Figure 14.6. If ( fr ) is a sequence of analytic functions deﬁned on Dr such that fr +1 is a direct analytic continuation of fr (r = 1, . . . , n − 1), then f n is an analytic continuation of f 1 from D1 to Dn . An analytic continuation that is not direct is an indirect analytic continuation. Unlike the direct case, we may obtain different results by using different sequences of domains, Figure 14.7. Indeed, we can even have a sequence of overlapping domains for which Dn = D1 , but f n ± = f 1, so that on returning to our starting point we end up with a different function from the original one. We illustrate this possibility by an example in the next section. But ﬁrst, we apply the above ideas to deﬁne a broader concept of an analytic function.

296

Analytic Continuation

Figure 14.7 Sequences of overlapping domains that may lead to different analytic continuations.

14.3.3

Complete Analytic Functions If f is analytic in a domain D we call the pair ( f , D) a function element. We deﬁne a relation ∼ on the set of all function elements by ( f1 , D1 ) ∼ ( f2, D2) if f2 is an analytic continuation of f1 from D1 to D2. Since indirect continuations are allowed, it follows easily that ∼ is an equivalence relation. A complete analytic function is an equivalence class under ∼ of function elements. In other words, a complete analytic function in the new sense is an analytic function in the old sense, together with all of its analytic continuations. It is clearly more convenient to have the analytic continuations ‘built in’ than to have to ﬁt them together as the occasion warrants. If there exist function elements ( f1, D1 and ( f2 , D2 ) of the complete analytic function F such that for some z ∈ D1 ∩ D2 we have f1 (z) ± = f2 (z), we say that F is multiform. If not, then F is uniform. All the examples considered so far in this chapter are uniform, but we give examples of multiform functions in the next section. A multiform F is a formal version of the classical idea of a ‘multivalued function’, with the advantage that it is broken down into pieces on each of which it is a genuine single-valued function. The multiformity arises because of how these pieces ﬁt together. A geometric approach to this leads to the concept of a Riemann surface, discussed informally in Section 14.5.

14.4

Multiform Functions A simple instance of a multiform function is f (z)

If z = rei θ then we can choose for

= √z

√z either

rei θ/2

or

reiθ/2+π

14.4 Multiform Functions

297

where r is real and positive. With the old concept of an analytic function, we must choose one of these arbitrarily, and then f (z) is analytic only if we make a cut in the complex plane. From the present viewpoint, we can do better. We spell the method out in considerable detail – just this once. We introduce four domains:

= {z ∈ C : re z > 0} H2 = {z ∈ C : im z > 0} H3 = {z ∈ C : re z < 0} H4 = {z ∈ C : im z < 0}

H1

which are the open half-planes to the right, top, left, and bottom of the complex plane. Let z = reiθ where r > 0, θ ∈ [−π , π ]. Deﬁne f1(z) f2(z) f3(z) f4(z) f5(z) f6(z) f7(z) f8(z)

= = = = = = = =

√reiθ/2 √reiθ/2 º √ iθ/2 √re i(θ/2+π ) √rei(reθ/2+π ) √rei(θ/2+π ) √rei(θ/2+π ) º √ i(θ/2+π ) √re iθ/ 2 √reiθ/re2

for for for for for for for for for for

z ∈ D1 = H1 z ∈ D2 = H2 z ∈ D3 = H3 , im z ≥ 0 z ∈ D3 = H3 , im z < 0 z ∈ D4 = H4 z ∈ D5 = H1 z ∈ D6 = H2 z ∈ D7 = H3 , im z ≥ 0 z ∈ D7 = H3 , im z < 0 z ∈ D8 = H4

Each of these eight functions is analytic in its domain of deﬁnition; this is why for f3 √ and f7 we have to change our choice of z as we cross the imaginary axis, because otherwise our choice −π < θ ≤ π would introduce a discontinuity. Further, fr +1 is a direct analytic continuation of fr (r = 1, . . . , 7) and f1 is a direct analytic continuation of f 8. For each z ∈ C \ {0} the values fr (z), where deﬁned, are one or other of the two possible square roots. Further, for each r, the function fr (z) takes one of the two values and fr +4 takes the other value, interpreting r + 4 modulo 8 (and replacing 0 by 8). Thus we have deﬁned a multiform function, taking two values at each z ± = 0. Going round the origin once, from D1 to D2 to D3 to D4 to D5 = D1, we reach a different value of f (z) from the original one. However, in this case, going round a second time returns √ us to the original value. This last effect is a special feature of z, as the next example shows.

14.4.1

The Logarithm as a Multiform Function Example 14.5. An immensely important multiform function is

f (z) = log z Its multiformity was Euler’s great discovery, arising from the Bernoulli–Leibniz controversy mentioned in Chapter 0. In terms of the present discussion, we introduce domains

298

Analytic Continuation

D4k +r = Hr

(k

∈ Z , r = 1, 2, 3, 4)

with the Hr as above. To avoid the kind of two-piece deﬁnition that occurred for f3 and f7 above, we proceed as follows. For z ∈ Dn and z = reiθ where n−2 π 2

we deﬁne fn(z)

< θ ≤ 2n π

= log r + iθ

Then fn is analytic on Dn. On D1 , f1 (z) = Log z the principal value of the logarithm. On D5, f5 (z) = Log z + 2π i On D4k +1,

f4k+1 (z) = Log z + 2kπ i

It is not hard to check that fs+1 is a direct analytic continuation of fs from Ds +1 to Ds for all s ∈ Z. For each r = 1, 2, 3, 4 and k ∈ Z the values of f4k+r (z) (z ∈ Hr ) give all the inﬁnitely many possible values log | z| + (2kπ

+ arg z)i

of the logarithm.

14.4.2

Singularities We can now give a general deﬁnition of a singularity. If no analytic continuation of f can be deﬁned at a point z0 then we say that z0 is a singularity of the corresponding complete analytic function. We have met several kinds of singularity before: poles, isolated essential singularities, natural boundaries. Removable singularities are not singularities at all according to the new deﬁnition, because ‘ﬁlling in’ the missing point is a form of analytic continuation. √ For z and log z we encounter a new kind of singularity, called a branch point. Analytic continuation round such a point leads to a change in value. Notice that for √ f (z) = z there is even a natural deﬁnition of f (z) at the branch point itself, namely 0, but f is not analytic there. This is not the case for the logarithm. Multiform functions make an appearance in contour integration whenever different choices of path from z0 to z1 leads to different values of the integral F(z1)

=

»

z1

z0

f (z) dz

(as may occur, for instance, if f has a pole in the region between the two paths). Hence quite nice singularities of f , such as poles, give rise to much nastier singularities of F,

14.5 Riemann Surfaces

299

namely branch points. This can occur even when f is uniform: for instance, 1/z is uniform but its integral log z is multiform, and the pole of f at 0 has become a branch point of F.

14.5

Riemann Surfaces Riemann was a great geometer, and he invented a geometric way to envisage multiform functions, which is much more intuitive than an equivalence class of function elements. His idea is to replace C by a more complicated ‘Riemann surface’. Roughly speaking, the idea is to ‘glue together’ the domains of function elements at overlaps where the functions agree. We sketch this idea in general later; our immediate aim is to describe a few key examples.

14.5.1

Riemann Surface for the Logarithm In the case of the logarithm we can describe the construction informally in the following terms, which should not be subjected to too deep scrutiny of a logic-chopping kind. We are not attempting a rigorous deﬁnition at this point; the informal description, though it may sound far-fetched, is in fact capable of being given a rigorous rendering, described at the end of this section. Consider a collection of copies Ck of C, one for each k ∈ Z. Slit each Ck along the negative real axis from 0 to −∞. For each k join the top left-hand quadrant of Ck to the bottom left-hand quadrant of Ck+1 along the slit. The resulting surface, Figure 14.8, resembles a spiral staircase or a concertina, with the planes stacked on top of each other in order relative to k, and with a continuous spiral path from any Ck to any other Cl going up the ‘steps’ where the slits were joined. These planes Ck are the sheets of the Riemann surface, and it is convenient to imagine the whole collection to be stacked on top of C as shown in Figure 14.8. We deﬁne the logarithm for points on the Riemann surface by log |ˆz| + (2kπ

Figure 14.8 Riemann surface for the logarithm.

+ arg z)i

300

Analytic Continuation

Figure 14.9 A chain of overlapping domains that closes up in

transferred to the Riemann surface.

C no longer closes up when it is

Figure 14.10 Riemann surface for the square root – schematic of how the two sheets join.

where ˆz is the point of Ck lying directly above the point z ∈ C. This is a single-valued function on the Riemann surface, and it is continuous in the sense that the values join up correctly across the cuts. (In a similar sense we can even say it is differentiable: the derivatives also join up correctly across the cuts.) Now we describe how to relate analytic continuation in C to the corresponding operation on the Riemann surface. Corresponding to a closed chain of domains D1, . . . , Dn = D1 ⊆ C, we have D³1, . . . , D³n on the Riemann surface. In C the values of the functions f1 , . . . , fn agree on the relevant overlaps D1 ∩ D2 , . . . , Dn−1 ∩ Dn, but because of the multiformity they do not agree on D1 and Dn. In contrast, on the Riemann surface we ﬁnd that if we make D³1 and D³2 overlap, then D³2 and D³3 , and so on, then by the time we get to D³n it no longer coincides with D³1 because we have moved one level up the Riemann surface, see Figure 14.9. Hence D³1 and D³n no longer overlap, and the difference between f1 and f n is only to be expected. The multiform nature of analytic continuation is automatically taken care of by the multiplicity of the sheets of the Riemann surface and how they join together. Even the nature of the singularity at 0 is apparent from the geometry of the surface, because here all the levels meet.

14.5.2

Riemann Surface for the Square Root

Similarly for z we obtain the Riemann surface by taking two copies C1 and C2 of C, slit from 0 to −∞. Then join the top left-hand quadrant of C1 to the bottom lefthand quadrant of C2 along the slit; and similarly join the top left-hand quadrant of C2 to the bottom left-hand quadrant of C1 . (To do this in three-dimensional space requires allowing the surface to intersect itself, but there is no conceptual problem here, see Figure 14.10.) Again the phenomena noted above are clear from the geometry of the Riemann surface: the two values of f (z), the fact that going round the the branch point at the origin once changes the value but going round twice does not.

14.5 Riemann Surfaces

301

Figure 14.11 Gluing successive domains to form the Riemann surface of the logarithm.

14.5.3

Constructing a General Riemann Surface by Gluing Obviously we cannot rely on such ad hoc methods to construct the Riemann surface in general. We can approach the general method through the same example of the logarithm, but building up the surface in a different way. Consider the sequence of half-planes Ds and functions fs used for analytic continuation above. Suppose we take half-planes D³s corresponding to the Ds , but make them pairwise disjoint. (The Ds are not disjoint; for instance D1 = D5 and so on.) Then we ‘glue’ the D³s together in the following way: Ds and Ds +1 overlap on a quadrant in which fs = fs +1, so we glue together the corresponding quadrants D³s and D³s +1 where they overlap. Then D³2 glues on top of D³1 at the top right-hand quadrant; D³3 glues on to the top left of D³2; D³4 glues on to the bottom left of D³3; and D³5 glues on to the bottom right of D³4. at this point D³5 is lying directly over D³1, but we do not glue them together since f1 and f5 are different, Figure 14.11. Continuing with D³6, D³7 , . . . (and in the downward direction D³0, D³−1, D³− 2, . . .) we build up the spiral staircase Riemann surface again. The general method for constructing a Riemann surface of a complete analytic function F follows the same lines. Recall that F is an equivalence class of pairs ( f , D) where f is analytic on a domain D and ( f1 , D1 ) ∼ ( f2 , D2) if f1 , f2 are equal on the non-empty set D1 ∩ D2 . We take disjoint copies D³λ of all the domains Dλ occurring in the pairs ( fλ , Dλ ) that belong to F. If Dλ ∩ Dµ ± = ∅ and fλ = fµ on it, we glue D³λ to D³µ at the points corresponding to the overlap Dλ ∩ Dµ . We make this description more rigorous. First, the question of ‘disjoint copies’ D³λ of Dλ . One way to achieve this is to deﬁne D³λ

= Dλ × {λ} Then if λ ± = µ it is clear that D³λ ∩ D³µ = ∅, and there is a natural bijection jλ : Dλ → D³λ deﬁned by jλ (s)

= (s, λ)

(s ∈ Dλ )

Now for that ‘gluing’. This is accomplished by a simple trick, using yet another equivalence relation, but it requires a certain amount of sophistication to appreciate that it does what is required.

302

Analytic Continuation

Let

(zλ , λ ) ≈ (zµ , µ )

∈ Dλ, zµ ∈ Dµ, if: zλ = zµ f λ (zλ ) = fµ (zµ )

for zλ (i) (ii)

Then the set of equivalence classes under ≈ of points in the union of all the D³λ is deﬁned to be the Riemann surface of F. The equivalence relation ≈ acts as the glue. This deﬁnition is quite elegant, but might not be immediately appealing. The advantage of the Riemann surface construction is that we can now think of a multiform complex analytic function as a genuine function, deﬁned not on C but on the appropriate Riemann surface, rather than as an equivalence class of function elements. The Riemann surface gives a ‘global’ view of the function, instead of chopping it up into ‘local’ pieces. The last few sections of this chapter revisit previously discussed material to explore some of the insights that can be gained in this way.

14.6

Complex Powers So far we have considered powers za of a complex number z only for rational a, where zp/q is a qth root of zp . We now deﬁne it for arbitrary a ∈ C. We would like to do this so that the laws za+b = za zb and (za)b = zab, (zw)a = zawa remain valid. It turns out that we can ‘almost’ do this: however, za is in general multiform, and the formulas hold only if we choose appropriate values. Even when a is rational this problem already arises. A natural approach is to write z = rei θ

where log r obtain:

= elog r+iθ

(14.7)

∈ R, and let a = α + iβ . On the assumption that the above laws hold, we za

= (reiθ )a = ea(log r+iθ ) = e(α+iβ)(log r+iθ ) = eα log r−βθ ei(β log r+αθ )

We therefore deﬁne za by (14.8). This makes sense for all z amounts to requiring that za

= ea log z

(14.8)

±= 0, and for all a ∈ C. It (14.9)

which is an equally natural way to deﬁne za. It follows that in any domain for which we may deﬁne a unique branch of log (such as the cut plane Cρ for any ρ ) we may also deﬁne za as a single-valued differentiable function. In general, however, since log is multiform, so is za. To see what the possibilities are, choose a particular value θ0 for θ (the obvious choice is the principal value of arg z). Then the possible θ that make (14.7) hold are of the form

303

14.6 Complex Powers

θ = θ0 + 2nπ

(n ∈ Z )

(14.10)

Write (za )n

= ea(log r+i(θ +2nπ )) 0

which is the ‘nth branch’ of za , obtained by substituting θ from (14.10) into (14.8). The question is now: how does (za)n depend on n? From (14.8) we have (za )n

= (e−2nπβ e2nπ iα )(za )0

(14.11)

where (za) 0 is the 0th branch. Using (14.11) we can answer the above question. There are three cases: (a) If β ± = 0 then za has inﬁnitely many values, one for each n, because

|(za)n /(za)0 | = (e−2nπβ )n which takes distinct values for distinct n. The Riemann surface for za in this case has the same form as that for the logarithm: an inﬁnite ‘spiral staircase’ on whose nth layer the function is given by the branch (za)n . (b) If β ± = 0, so a = α ∈ R, and if further α is irrational, then za = zα is again inﬁnitely many-valued, with distinct branch values for distinct n. For if (zα) m = (zα )n then e2m π iα = e2nπ i α , so e2π i(m −n)α = 1. This implies that (m − n)α ∈ Z by Section 5.7, which implies that m = n since α is irrational. The Riemann surface for zα is the same as for case (a), but unlike case (a), the moduli of distinct branches are equal. Only the arguments vary. (c) If β ± = 0, so a = α ∈ R, and if further α is rational, then we can write α = p/q in √ lowest terms, where p, q ∈ Z. Now zα = q zp. Following the discussion in (b), two branches give the same values if and only if (m − n)

p q

∈Z

which happens if and only if q divides m − n, because p and q have no common factor greater than 1. It follows that the value of (zα)n depends only on n mod (q). The branches (zα )0 , (zα )1, . . . , (zα )q−1 are distinct, but the qth branch repeats: (zα )q

= (zα )0

and repetitions continue thereafter. The Riemann surface, of course, is a q-sheeted spiral, with its top sheet glued to its bottom one, as in our discussion of z1/ 2 but with q sheets instead of 2. Figure 14.12 illustrates the case q = 5.

304

Analytic Continuation

Figure 14.12 Section through the Riemann surface for z p/ q when q

glue together.

= 5, showing how the layers

Figure 14.13 Conformal map of a half-plane to a wedge.

Figure 14.14 How the conformal map of a half-plane to a wedge transforms streamlines.

14.7

Conformal Maps Using Multiform Functions Conformal mapping with multiform functions requires careful attention to how values are speciﬁed; again, the simplest way to keep track is to put everything on a Riemann surface. Here we consider only one case, of practical importance: the map z → zα for real α ≥ 0. We assume the domain is the cut plane Cπ , on which the map may be deﬁned to be single-valued. The great virtue of this map is that it transforms a half-plane into a wedge, as in Figure 14.13. The vertex angle of the wedge is απ , so we have to assume α < 2 to prevent the image overlapping itself. The map is conformal except at the origin. For example, uniform ﬂuid ﬂow, with streamlines re z = constant, transforms to the ﬂow round a corner of angle απ , as in Figure 14.14. If we take α = 1/2, the corner is right-angled. Then z = x + iy and z1/ 2 = w = u + iv. The ﬂow-lines are x = constant. Now z = w2 so x = u2 − v2, y = 2uv. The streamlines in the w-plane are given by u2 −v2 = constant; they are branches of rectangular hyperbolas.

14.8 Contour Integration of Multiform Functions

305

Figure 14.15 Transferring a contour to a Riemann surface.

In combination with other ‘standard’ conformal maps, zα provides a useful weapon for the applied mathematician.

14.8

Contour Integration of Multiform Functions ¼

It is conventional to interpret a contour integral γ f of a multiform f by choosing the values of f (z) so that they vary continuously along the contour. (This still leaves an arbitrary initial choice to be made, which must also be speciﬁed.) A more civilised approach is to break the contour γ into a sum γ = γ1 + · · · + γn such that: (i) Each γj lies inside a domain Dj on which f may be deﬁned as a single-valued function. (ii) On each Dj we choose a branch of f that makes the values agree where γj and γj+1 join. In terms of Riemann surfaces, we can interpret this process as the deﬁnition of a contour integral when the contour lies on the Riemann surface. The way the contour wanders up and down the staircases on the surface automatically takes care of the choice of values of the multiform function f (z), see Figure 14.15. In practice this is all more straightforward than it may sound, and the computations are no harder than the uniform case – provided you use your head.

Example 14.6. Let

Find

γ be the contour γ (t) = (1 + t)eit »

γ

(t

∈ [0, 6π ])

z1/5dz

Method 1 (stupid): Deﬁning z1/5 on the contour by (γ (t)) 1/ 5 vary continuously with t. By Theorem 6.6

= (1 + t)1/5 eit/5 makes it

306

Analytic Continuation

θ

Table 14.1 Choice of .

domain D1 D2 D3 D4 D5 D6

θ chosen in interval [−π/4, 5 π/4] [3π/ 4, 9π/4] [7π/ 4, 13π/4] [11π/4, 17 π/4] [15π/4, 21 π/4] [19π/4, 25 π/4]

Figure 14.16 Six overlapping domains.

»

γ

f (z)dz =

=

» 6π 0

» 6π 0

( f (γ (t)) γ ³(t) dt (1 + t)1/ 5eit /5 (ieit + iteit

+ eit ) dt

which leads the hopeful calculator along some fascinating byways inhabited by integrals such as

» 6π 0

t(1 + t)1/5 cos(6t /5)dt

A wise person knows when to cut their losses. Method 2 (sound but pedestrian): Cover the contour γ by domains in which z1/5 may be given a unique value for each z, which is differentiable. For example, we can ﬁnd six domains D1 , . . . , D6, one for each of the intervals AB, BC, CD, DE, EF, FG in Figure 14.16. To make the values match on overlaps we choose the deﬁnition of z1/5 as follows. Write z = reiθ where the choice of θ is as in Table 14.1. On each domain choose the branch of z1/5 given by r1/5 eiθ/5 with θ in the range stated in the table. These choices make θ agree on overlaps.

14.8 Contour Integration of Multiform Functions

Break up the integral as

»

γ

»

=

AB

+

» BC

+

» CD

+

» DE

+

» EF

+

307

» FG

In each domain Dj the function z1/5 has an antiderivative 65 z6/ 5 where z6/5 is deﬁned using the corresponding branch z6/ 5 = r6/ 5e6iθ/5 . By Theorem 6.33, if PQ is any of the arcs AB, BC, and so on, then

»

PQ

z1/5 dz = 65 Q6/5 − 65 P6/5

When we add the six expressions of this type, all terms but two cancel, so the integral is 5 6/5 6G

− 65 A6/5 Here A = γ (0) = 1, G = γ (6π ) = (1 + 6π )e6iπ . Evaluate the 6/5th powers according to the above prescription:

A6/ 5 = (1 · ei0 )6/5 G

6/ 5

= (1 + 6π )e

=1

6π i 6/5

)

= (1 + 6π ) 6/5e36iπ/5 = −(1 + 6π )6/5 eiπ/5

Therefore the integral is equal to

− 65 [(1 + 6π )6/5 eiπ/5 + 1] Method 3 (slicker): In terms of the parameter t we can deﬁne the required branch of z1/5 to vary continuously along the contour γ if we put z1/ 5 = ( γ (t))1/ 5 = (1 + t)1/ 5eit/ 5

There is a local antiderivative 65 z6/5 , whose value varies continuously along the path γ if we choose branches so that (γ (t))6/ 5 = (1 + t)6/ 5e6it/ 5

Now pave γ by a ﬁnite number of domains, on each of which z1/5 may be rendered single-valued, choosing the branches to agree with those already chosen on the path. As above, the integral may be expanded as a sum of terms of the form 5 [(1 6

+ tj+1 )6/5 e6it + /5 − (1 + tj )6/5 e6it /5 ] j 1

j

where the t j subdivide the interval over which t runs. All but two terms cancel, leaving the same answer as before. The advantage of this method is that by choosing branches in an obvious way for γ (t) we may leave the prescription of the domains, and the branches on those domains, implicit. Method 4 (smart): Let the Riemann surface do the work. The above analysis easily generalises to give a version of Theorem 6.33 for multiform functions, integrated along

308

Analytic Continuation

a contour in the Riemann surface. Given a global antiderivative F for f on the Riemann surface, we have

»

γ

f

= F(z1) − F(z0)

where z0 is the initial point, z1 the ﬁnal point, and the branches are chosen with reference to the end points of γ on the surface, as in Figure 14.15. This gives the result almost immediately on noting that γ winds three times anticlockwise, so the 6 /5th power winds 3 · 6/5 = 18 /5 times anticlockwise, giving for F(z1 ) the argument 2π · 18/5 − 36π/5, if we choose argument 0 for z0.

We leave it as an exercise to formulate and prove generalisations of Cauchy’s Theorem for contour integrals on the Riemann surface of a multiform function. For those who prefer not to use the insight that Riemann surfaces provide, however, we recommend Method 3 above as a relatively direct and simple technique.

Example 14.7. Example 14.6 is of course a bit artiﬁcial, and its main interest is pedagogical. A more typical example of integrals likely to be encountered in practice is » ∞ xa dx (a R, 0 a 1) 1 x2 0

+

<
1. Then the singularities inside γ are z = ²i. We calculate the residues at these points as follows: At z = i the residue is

At z

1 iaπ/2 e α = lim za /(z + i) = 2π i z →i

= −i the residue is

1 ia3π/2 β = zlim za/(z − i) = e →−i 2π i Let the circle of radius R be Theorem implies that

»

R

xa

ρ 1+x

dx + 2

»

za

γ1 1 + z

γ1, and that of radius ρ be γ2. Then Cauchy’s Residue

dz + 2

» ρ 2π i a (xe ) 1 + x2

R

dx +

»

za

−γ2 1 + z2

dz = 2π i(α+β ) (14.12)

Note that in the third integral we must choose the branch of za corresponding to argument 2π , to retain continuity. Now let ρ → 0 and R → ∞. Easy estimates show that the second and fourth integrals tend to zero. The ﬁrst tends to

» ∞ 0

and the third to 2π ia

−e Therefore (1 − e

2π ia

)

» ∞ 0

xa

xa

1 + x2

» ∞ 0

1 + x2

dx = 2π i

0

xa

1+

x2

xa

1 + x2

After some manipulation, this leads to

» ∞

dx

dx =

±

1

dx

2π i

iaπ/2

e

1

ia3π/2

− 2π i e

²

π

2 cos(aπ/2)

Yes, but . . . did you spot the nifty footwork? The chosen contour does not actually lie inside the domain C0 . Nonetheless, the result is correct. There are several ways to justify the answer rigorously. They include: (1) Replace γ by a contour that starts just above ρ , say at ρ + iε, runs horizontally to just above R at R + iε , circles γ1 to just below R at R − iε, runs back to below

310

Analytic Continuation

Figure 14.18 Modiﬁed contour remains inside

Figure 14.19 Lifting

C0 .

γ to the Riemann surface.

ρ at ρ − iε, and the goes back round γ2 as in Figure 14.18. Now, as ρ → 0 and R → ∞, make the width 2ε of the channel tend to zero and use continuity arguments. (2) The choice of C0 causes problems, so reject it. Work in two overlapping domains, say C−π/4 and Cπ/4, and switch between them when deﬁning the second, third, and fourth integrals in (14.12). (3) Put it all on a Riemann surface. The contour γ then winds round the staircase and up one step, then winds back down one step to get back to the start. Everything generalises to such contours, because they are obviously boundaries of rectangles in the Riemann surface, Figure 14.19. This argument is an example of the ‘homology’ version of Cauchy’s Theorem, when generalised to Riemann surfaces. We discuss homology in Chapter 16, but do not extend the ideas to Riemann surfaces since this would take us too far aﬁeld. These three approaches justify the computational method rigorously. Of course, when performing the calculations, it is not necessary to trot out such a justiﬁcation every time – provided you understand how it would go. Everything works ﬁne as long as you follow the golden rule: make the multiform function vary continuously as you walk round the contour.

14.9 Exercises

14.9

311

Exercises 1. Deﬁne three power series by a(z)

= 1 + z + z2 + z3 + · · · =

∞ µ n=0

zn

∞ µ b(z) = i − (z − i − 1) − i(z − i − 1) + (z − i − 1) + · · · = i n+1(z − i − 1)n 2

c(z)

= −1 + (z − 2) − (z − 2)2 + · · · =

3

∞ µ n=0

n=0

(−1)n+1 (z − 2)n

Find their discs of convergence and sketch them. Prove that a(z) = b(z) on the overlap of their discs of convergence, and similarly that b(z) = c(z) on the overlap of their discs of convergence. Do the discs of convergence of a(z) and c(z) intersect?

∞ bn(z − z2 )n be two power series, n 2. Let f (z) = ∞ n=0 a n(z − z1) and g(z) = n=0 and assume that their discs of convergence have non-empty intersection. Prove that g is a direct analytic continuation of f if and only if there exists z0 in both discs of convergence such that for m = 0, 1, 2, 3, . . . the equations m µ n=0

n(n − 1) · · · (n − m)[an(z0 − z1 )n−m − bn (z0 − z2)n−m ]

=0

hold. 3. A certain function has singularities at precisely the points z such that re z and im z are integers. You are given its Taylor expansion around the point 21 + 12 i. What is the smallest number of stages needed to ﬁnd an indirect analytic continuation, by power series, to a domain that contains 27 + 25 i? 4. Show that the functions deﬁned by n (i) f (z) = 1 + z + z2 + z4 + z8 + · · · + z2 + · · · n (ii) g(z) = 1 + z + z3 + z9 + z27 + · · · + z3 + · · · have natural boundaries at | z| = 1.

= ∑ anzn has radius of convergence 1. Set w = w − w2 + w 3 − w4 + · · · z= 1+w ∑ and transform f (z) to a power series in w, say F(w) = bn wn . Prove that this latter 1 power series has radius of convergence ≥ 2 , and that if −1 is a singular point of f ,

5. Suppose that f (z)

the radius of convergence is exactly 21 . 6. Show that the series

∞ µ n=1

[(1 − zn+1) −1 − (1 − zn )−1]

converges when |z| > 1 or | z| < 1, but that the two functions so represented are not analytic continuations of each other.

312

Analytic Continuation

7. Suppose that f and g are deﬁned, and have no singularities, on the whole of Deﬁne

²(z) =

∞ ± 1 − zn µ n=1

1 + zn

1 − zn−1 1 + zn−1

C.

²

Show that

+ g(z)) + 21 ²(z)( f (z) − g(z)) is equal to f (z) when | z| < 1, but to g(z) when | z| > 1. √ Let f (z) be the (multiform) function z z. Show that at z = 1 ( f (z) 2

8.

0 there exists a ﬁrst derivative that is the same for all branches, but a ﬁnite second derivative does not exist. What about z2 log z?

9. Describe the Riemann surfaces of the multiform functions: √7 (i) √ z + 43 (ii) 1 − z3 (iii) cos−1 z (iv) tan−1 z 10. Describe the Riemann surface of [(z − 1)(z − 2)−2 ]1/3 + (z − 3)1/2 11. Let ω ∈ C. Show that the values of 1ω form a subgroup Uω of the multiplicative group of non-zero complex numbers. Show that Uω is cyclic of order q if ω is a rational number p/q in lowest terms; otherwise, Uω is inﬁnite cyclic. Show that Uω is contained in the unit circle if and only if ω is real; lies on the positive real axis if and only if re ω is an integer; and otherwise lies on a logarithmic spiral parametrised by t ∈ R in the form eαt for some ﬁxed complex number α. 12. Show that the function

f (z) = e−iπ/8 z − i deﬁnes a conformal map from the domain in Figure 14.20 (left) to the upper halfplane.

Figure 14.20 Left: Domain for Exercise 12. Right: Domain for Exercise 13.

14.9 Exercises

313

Figure 14.21 Sector occurring in Exercise 14.

Figure 14.22 Contour for Exercise 16.

13. Show that the function

²2/ 3 ± − iπ/3 z + 1 f (z) = e z −1

deﬁnes a conformal map from the domain in Figure 14.20 (right) to the upper halfplane.

14. Find the image of the sector −π/n the conformal transformation

< arg z < π/n, |z| < 1 (Figure 14.21) under

f (z) = z(1 + zn)2/n where n is a positive integer.

15. Let γ (t) = eit , (t

√ ¼ ∈ [0, 4π ]). Calculate γ √z dz where at t = 0 we take 1 = 1.

16. Using the contour in Figure 14.22 show that

» ∞ −k x 0

1+x

dx = π cosec kπ (0 < k < 1)

17. Show by contour integration that

»

1

0

(1 +

dx

ax2 )

1−

x2

= √π

2 1+a

(a > 0)

18. Using the contour of Exercise 16, show that

» ∞ 0

π (1 − a) xa dx = 2 2 (1 + x ) 4 cos 12 a

(−1 < a < 3)

314

Analytic Continuation

Figure 14.23 Schwartz Reﬂection Principle.

19. Let γ (t)

= (1 + 21 cos t + i sin t)5, (t ∈ [0, 2π ]). Find » √ z99 log z − z + (z − 1 ) 5/ 17dz 4

γ

20. Describe the Riemann surface of the function f (z) =

½

z + z2 + z4 + z8 + · · · + z2

n

+ ···

21. Schwartz Reﬂection Principle. Let U be a domain in C that is symmetric about the real axis (that is, if z ∈ U then ¯z ∈ U). Let U+

= {z ∈ U : im z > 0}

U0

= {z ∈ U : im z = 0}

U − = {z ∈ U : im z < 0} as in Figure 14.23. Suppose that f : U + ∪ U0

→C

is continuous, analytic on U + , and takes real values for z ∈ U 0. Then there is an analytic function F : U → C such that F(z) = f (z) for all z ∈ U+ ∪ U 0. (Hint: deﬁne F(z) = f (¯z) for z ∈ U − , and use Morera’s Theorem.)

15

Inﬁnitesimals in Real and Complex Analysis

Historically, calculus began with the study of variable quantities represented as curves in the plane. These variables could become ‘arbitrarily small’ or ‘inﬁnitesimal’. They were used to compute derivatives and integrals, and to formulate differential equations modelling systems that change dynamically. However, the precise meaning of an inﬁnitesimal caused controversy. At least two different concepts existed: a ‘number’ (of some kind) that is smaller than any non-zero number, or a variable number that can become smaller than any positive number. The two were often confused. Moreover, it is not easy to formalise the idea of a ‘variable number’. (If it varies, how can it be a number?) Because of these difﬁculties, inﬁnitesimals, in either sense, were eventually displaced by the logical epsilon-delta formulation of Weierstrass in the late nineteenth century. The real numbers form a ‘complete ordered ﬁeld’, and therefore do not contain inﬁnitesimals, so inﬁnitesimals were banished from formal mathematical analysis. Meanwhile, applied mathematicians and theoretical physicists continued to model the √ real world using ‘arbitrarily small’ quantities. The familiar formula T = 2π L /g for the period of a pendulum, derived using the approximation of simple harmonic motion, is a familiar example. Usually described as ‘valid for small amplitude swings’, this formula is actually incorrect for any non-zero amplitude. But is is also a very good approximation for sufﬁciently small amplitudes, and more accurate formulas cannot be derived or expressed using elementary functions. This procedure is typically justiﬁed by letting the approximations concerned become arbitrarily small – the main traditional view of an inﬁnitesimal. Thus two parallel but apparently conﬂicting approaches to calculus arose: one with, and one without, inﬁnitesimals. Moreover, the tacit assumptions about inﬁnitesimals applied to two different concepts. In this chapter we reconcile these views, offering a formal approach to inﬁnitesimals that lets them be manipulated algebraically. Real inﬁnitesimals can be visualised on an extended number line, and their complex analogues can be visualised in an extended complex plane. Inﬁnitesimals are made visible to the human eye by a process of magniﬁcation, which is deﬁned algebraically and has a natural visual interpretation. The formal representation includes the idea of an inﬁnitesimal quantity as a process that tends to zero, and also as an element in an ordered ﬁeld extension K of the reals, and in the corresponding extension of C. We prove a simple structure theorem for such an extended system that any ﬁnite element x in the extended system (meaning | x| < r for

316

Inﬁnitesimals in Real and Complex Analysis

Figure 15.1 Leibniz’s view of dx, dy as components of the tangent.

some real number r) is uniquely of the form x + h, where x is a real or complex number and h is inﬁnitesimal or zero. This approach provides a formal theory that encompasses both the standard epsilondelta approach to analysis in pure mathematics, and the ‘applied’ approach in which inﬁnitesimals are thought of as variables that tend to zero.

15.1

Inﬁnitesimals Leibniz, Newton, and their contemporaries used various imaginative devices to deal with differentiation and integration. Leibniz’s ﬁrst publication on calculus speciﬁed the derivative dy/dx as the quotient of two ﬁnite numbers dy and dx. Figure 15.1 illustrates his procedure. Suppose that the tangent to a point P on the curve mets the horizontal axis at B. For any ﬁnite horizontal displacement dx, let dy be the corresponding vertical change of the tangent. Then the ratio dy/dx is the same as y/BX. In this interpretation, dx and dy do not represent inﬁnitesimals. They represent ﬁnite components of the tangent vector. When the slope of the tangent is m, we can take any ﬁnite number dx to be the horizontal component, and the vertical component is then dy = m dx The slope of the tangent to the curve at P is the quotient dy/dx. The devil lies in the detail. The deﬁnition makes sense only when the slope of the tangent is known, and to do that, it must be calculated. This is where inﬁnitesimals came in. Leibniz performed the calculation by imagining the graph to be a polygon with an inﬁnite number of inﬁnitesimally small sides. The tangent at a point was the prolongation of one of these sides, as in Figure 15.2, where the lengths of the sides are imagined to be arbitrarily small. He also conceived of different degrees of inﬁnitely small or inﬁnitely large quantities, by comparing the size of a ball to that of the Earth, or to the distance to the ﬁxed stars. This led him to imagine inﬁnitesimals of different orders of size. An inﬁnitesimal of second order is inﬁnitely smaller than an inﬁnitesimal of ﬁrst order, and so on. The reciprocal of an inﬁnitesimal of a given order is an inﬁnite element of the same order. For instance, if v is a ﬁrst order inﬁnitesimal, then v2 is a second order inﬁnitesimal and 1/v2 is an inﬁnite element of second order. Leibniz’s ideas were natural, in the practical

15.1 Inﬁnitesimals

317

Figure 15.2 The tangent as the extension of an inﬁnitesimal side.

sense that he imagined arbitrarily small or large quantities, and in the theoretical sense that they can be manipulated symbolically. Their logical status was less clear, and disputed. The philosopher George Berkeley, ridiculed inﬁnitesimals, claiming that it was beyond common sense to imagine quantities smaller than any sensible quantity, and even more nonsensical to imagine quantities that are smaller still. He referred to inﬁnitesimals sarcastically as ‘ghosts of departed quantities’, neatly confusing the two different interpretations. For generations, mathematicians disputed the precise meaning of this elusive concept, seeking to circumvent the logical difﬁculties. Cauchy succeeded, but his ideas were widely misunderstood, even today. He formalised the ‘variable quantity’ approach, deﬁning an inﬁnitesimal as a sequence α = (an) that tends to zero. He could then calculate f (x+α) for inﬁnitesimal α as the sequence ( f (x + an )). He also formulated the notion of continuity for a function in such terms: f (x) is continuous if, for any inﬁnitesimal α, the difference f (x+α) − f (x) = ( f (x +an) − f (x)) is also inﬁnitesimal. As he grew conﬁdent using inﬁnitesimals, Cauchy manipulated them in the same way as he manipulated variable quantities. He imagined points and lines after the style of the ancient Greeks, where a line is an entity that has ‘length but no breadth’ and a point is an entity that has ‘position but no size’. A line is an entity in its own right. A point can be marked on a line and a line can be drawn through a point. Furthermore, points can be constant, remaining in a ﬁxed position, or variable, including inﬁnitesimals as variables that tend to zero. What is less obvious is the nature of the number line itself. Not √ only is it possible to mark rational points on it, but there are also points such as 2 or π that arise in geometric constructions but are not rational. At the end of the nineteenth century, a solution was found by formulating the completeness property of the number line, so that it includes not only the rational numbers, but also the limits of convergent rational sequences. This notion of the reals cannot include inﬁnitesimals if they are deﬁned as ‘smaller than any positive real number’, because this concept is logically contradictory. A signiﬁcant change in interpretation then occurred. Instead of a line being something drawn physically by the stroke of a pen, or imagined theoretically as a Platonic entity with length but no breadth, it was re-conceptualised as the set of all real numbers – deﬁned, either as limits of convergent (that is, Cauchy) sequences of rational numbers, or as Dedekind cuts on the rationals [21]. Now every real number occupies a ﬁxed place on the number line. It cannot move around, or become ‘arbitrarily small’, so the formal

318

Inﬁnitesimals in Real and Complex Analysis

real number line cannot include inﬁnitesimals in Cauchy’s sense. Indeed, the notion of a ‘variable’ was reformulated as a ﬁxed but arbitrary member of some set. Movement was effectively banished from the fundamentals of mathematics. In consequence, the notion of an inﬁnitesimal was excluded from formal set-theoretic analysis. Yet the idea that quantities can vary, and become arbitrarily small, remains in the mental imagery of practising mathematicians, especially in applied mathematics. The recent arrival of dynamic computer graphics offers an entirely new practical way to visualise mathematical ideas. Greek geometers drew a curve as a mark in the sand; later generations drew it in ink on paper, or printed it in a book. Today we can represent a curve on a high-resolution visual display, and zoom in on part of the curve to magnify the image. Unlike the magniﬁcation of a picture in a textbook, where the thickness of the graph increases under a magnifying glass, we can now move our ﬁngers apart on a tablet computer to magnify the picture while the graph is redrawn at the same visual thickness. As we zoom in on a standard smooth graph such as y = x2 , y = ex , or y = sin x, the magniﬁed picture looks less and less curved. At successively higher magniﬁcations, a small portion of the graph looks more and more like a straight line, and the graph and the tangent become visually indistinguishable, see Figure 15.3. Now, under high magniﬁcation, a picture of a smooth graph magniﬁes locally to look like a straight line. This suggests a new way of imagining the tangent to a differentiable function: under high magniﬁcation, the graph looks less and less curved, until at suitably high magniﬁcation a small portion of the graph looks straight. The derivative is now visually the slope of the graph, and as we cast an eye along the graph, the derivative is the changing slope of the graph itself. This property can be visualised for combinations of standard polynomial, trigonometric, logarithmic, and exponential functions, but it needs to be formally deﬁned to reach modern standards of rigour, and suitably interpreted to apply to complex analysis and higher-dimensional vector calculus.

15.2

The Relationship Between Real and Complex Analysis The generalisation from the real to the complex case follows the same symbolic formulation, but the visual representations are different. In the real case, we can draw

Figure 15.3 Successive magniﬁcations of a smooth curve.

15.2 The Relationship Between Real and Complex Analysis

trace

319

images of trace

z point here

f

w = f(z)

z-plane

w-plane

w-plane w = f(z)

z-plane

z

a

γ

t

b

Figure 15.4 Top: Tracing a point in the z-plane to see its image move in the w-plane. Bottom: Visualising this in three dimensions by stacking planes.

the graph of a function f : D → R as a subset {(x, f (x)) ∈ R2 : x ∈ D}, but this cannot be directly generalised to the complex case because the graph is a subset of C × C, which requires four real dimensions. However, we can use alternative representations in two or three dimensions. For example, Figure 15.4 (top) represents the complex function w = f (z) by drawing parts of the z-plane and the w-plane side by side. If a ﬁnger starting at a point z traces a path in the z-plane, then the corresponding point w = f (z) moves around in the w-plane. This function can be visualised in three-dimensional space by placing the w-plane above the z-plane, with an arrow connecting z to f (z). This representation is similar to that used in Figure 2.9, but with an extra dimension in the z-plane. Then, as z moves in the z-plane along a path γ , the arrow moves along the image path f γ , tracing out the motion of w = f (z), Figure 15.4 (bottom). The movement of the ﬁnger travelling smoothly along a curve in the z-plane without stopping can be represented formally by a path γ : [a, b] → C, where γ is differentiable and γ ± (t) ² = 0 for t ∈ [a, b]. Recall from Deﬁnition 6.21 that we call such a path a regular curve. In this case, the tangent to γ at z0 = γ (t0) makes an angle arc (γ ± (t0 )) with the real axis. In Section 13.3, for an analytic function f , we distinguished between regular points z0 where f ± (z0) ² = 0 and critical points where f ± (z0 ) = 0. The behaviour of the transformed path f γ near w0 = f (z0 ) depends on this distinction. The transformed image path w = f (γ (t)) satisﬁes ( f γ )± (t0 ) = f ±(z0 )γ ± (t0 ), and if z0 is a regular point of f , that is f ± (z0) ² = 0, then the image path has a tangent at w = f (z0) as in Figure 15.5. The tangent makes an angle arc ( f ± (z0 )γ ± (t0 )) = arc ( f ± (z0 )) + arc (γ ± (t0 ))

320

Inﬁnitesimals in Real and Complex Analysis

Figure 15.5 Mapping a path

γ in D by an analytic function f where f ± (z0) ²= 0.

Figure 15.6 Tangent to curve in the w-plane.

with the real axis. The transformation f turns the tangent at a regular point through an angle arc( γ ±(t0)).

15.2.1

Critical Points In Section 13.3 we compared the geometry of a differentiable complex function f near a regular point and near a critical point. For a regular point z0 , the tangent to a regular curve γ through z0 has a corresponding tangent through f (z0).

= γ = + ∈− f (γ (t)) = sin(t + it) = sin(t) cos(it) + cos(t) sin(it) = sin(t) cosh(t) + cos(t) sinh(t)

Example 15.1. Consider the function f (z) sin z at the origin where f ± (0) The path (t) t it for t [ 1, 1] in the z-plane is transformed into

= cos 0 = 1.

in the w-plane, see Figure 15.6. At a critical point, f ± (z0) = 0, so τ collapses to a single point: τ (z0 + h) = w0. In Chapter 13 we noted that f need not be conformal at a critical point. In this case, a graph

15.2 The Relationship Between Real and Complex Analysis

Figure 15.7 Transforming the path

321

γ (t) = teiθ by the function w = z 2.

γ in the z-plane passing through z0 is transformed by an analytic function f in a subtler manner.

=

=

Example 15.2. Consider the function f (z) z2 at the origin where f ± (0) 0. The path (t) tei θ for t [ r, r] in the z-plane is transformed by f into the composition

γ =

∈−

f γ (t)

= t2 cos(2θ ) + it 2 sin(2θ ) in the w-plane. (Recall we now write f γ instead of f ◦γ , for brevity.) The original path γ in the z-plane starts at −reiθ and moves in a straight line through the origin to the point reiθ on the other side; the path f γ in the w-plane lies on the straight line at an angle 2θ to the real axis, starting at ( −r)2 e2iθ = r 2e2iθ moving through the origin and turning

back in the opposite direction to r2 e2iθ , see Figure 15.7. This apparently sudden turn in direction turns out to be a direct generalisation of the real case. If the real function f (x) = x2 is drawn with separate x- and y-axes, as x increases from −r to +r through the origin, the y-value x2 starts at r2 , moves down to the origin, and then turns back up again to the starting point. An alternative way to grasp what is happening is to represent the path γ : [a, b] → C by writing f γ (t) = u(t) + iv(t) and looking at the graph of (t, u(t), v(t)) in threedimensional (t, u, v)-space. In Figure 15.8 the w-plane is drawn horizontally and the t-axis vertically upwards. As t moves up the vertical axis from −r to r, (t, u(t), v(t)) moves along a path in three dimensions, with the tangent at the origin being vertical where its projection onto the w-plane is a single point. Looking down on the w-plane, the point (u(t), v(t)) moves from r2 e2iθ to the origin and turns back again. Imagining t as time, the velocity of the moving point on the path f is ( f γ )± (t) = 2te2iθ , and as t passes through the origin the velocity smoothly slows down to zero, and then reverses without any sudden change in speed.

More generally, if a point moves along any smooth path γ through a critical point in the z-plane, then the path f γ in the w-plane may turn back in the opposite direction. In three-dimensional (t, u, v)-space, however, it travels smoothly through the critical point. For example, for w = z2 , the path γ (t) = 1 − eiπ t (t ∈ [−1, 1]) is a circle radius 1, centre 1, which travels through the critical point at the origin in the z-plane. Meanwhile the path

322

Inﬁnitesimals in Real and Complex Analysis

Figure 15.8 The path f

γ in (t, u, v)-space.

Figure 15.9 Magnifying a path in the w-plane through a critical point of f.

in the w-plane passes through the origin as in Figure 15.9. Magnifying the picture in the w-plane near the origin reveals the path moving in and out in what looks like a straight line.

15.3

Interpreting Power Series Tending to Zero as Inﬁnitesimals Using the idea that an inﬁnitesimal quantity is a variable that tends to zero, the variable part of a power series can be interpreted as an inﬁnitesimal. Moreover, it has a special property that relates to Leibniz’s idea that inﬁnitesimals have an order of size. Recall from Deﬁnition 13.10 that an analytic function f : D → C is of order n at z0 ∈ D (where n ≥ 1) if f (n) (z0) ² = 0, and f (k)(z0 ) = 0 for 1 ≤ k < n. By (13.6) the Taylor series about z0 then has the form f (z0 + h) = f (z0) + hnf (n) (z0) /n! + · · · which is a constant f (z0 ) plus an inﬁnitesimal of order n. So the behaviour of f (z) near a critical point z0 is given by the order of the variable part of the analytic function at the point. The deﬁnition applies to regular points where n = 1, and to critical points where n > 1, to give a single theory in all cases.

15.4 Real Inﬁnitesimals as Variable Points on a Number Line

323

Using polar coordinates, we observed that if h = reiθ , where h is small, then f transforms a small region of the z-plane near z0 to the w-plane near w0 = f (z0). The radius r scales to krn , and the original angle θ is multiplied by n and rotates to nθ + α, as we saw in Chapter 13, Figure 13.6. Moreover, we noted that a further rotation of the line from z0 to z0 + h in the z-plane through φ rotates the corresponding line in the w-plane to n( θ + φ ) + α, as in Figure 13.7. The case of a straight line passing through z0 can now be considered as two halflines where φ = π . The image in the w-plane (for suitably small h) is then made up of two half-lines, one at an angle θ to the real axis, the other at an angle θ + π . Under the transformation f , the angle between them is increased to nπ . For n odd, this gives two half-lines in opposite directions and, for n even, it gives two half-lines in the same direction. This is a natural generalisation of the real case where, for n odd, as x increases through zero, so does y, while for n even, the graph of y = xn has a minimum at x = 0, and as x passes through the origin, y travels down to zero and turns back up again. The language used in the preceding discussion has been expressed in terms of a variable h that is ‘sufﬁciently small’, linking to the Leibniz notion of inﬁnitesimals of various orders of size and the dynamic idea of ‘arbitrarily small’ quantities in applied mathematics. We now introduce a formal set-theoretic deﬁnition of inﬁnitesimals and prove a structure theorem that reveals how they can be visualised on a number line or in a complex plane.

15.4

Real Inﬁnitesimals as Variable Points on a Number Line The structure of the human visual system makes it natural for us to imagine points moving along a line. Given a vertical line in the real plane of the form x = v, consider where various polynomials intersect this vertical line as it is moved to the left or right, Figure 15.10. A constant function y = k always meets the line at the same height k, but the line y = x meets the line at a height v, and as the line moves, v varies. As v gets smaller, the variable point v moves below any ﬁxed positive constant k. In this sense, as v tends to zero, the variable point becomes less than k for any real number k – one

Figure 15.10 Inﬁnitesimals as variable points on a line.

324

Inﬁnitesimals in Real and Complex Analysis

interpretation of ‘inﬁnitesimal’. In the same sense, the variable point v2 is smaller still – inﬁnitesimal compared to v. The vertical line x = v can now be seen to have two kinds of points. Some are familiar constant points that remain in the same position, while others are variable, and can become arbitrarily small. Others can become even smaller.

15.5

Inﬁnitesimals as Elements of an Ordered Field The plan now is to extend the real numbers R to a larger ordered ﬁeld K that contains inﬁnitesimals, and to extend real functions f so that we can calculate f (x) for x ∈ K. Taking a cue from the previous section, we might begin with the ﬁeld R(v) of rational functions in v, a0 + a1 v + · · · + an vn b0 + b1v + · · · + bm vm

where a0 , a1, . . . , an, b0 , b1, . . . , bm ∈ R and bm ² = 0, and give it the structure of an ordered ﬁeld by deﬁning v to satisfy 0 < v < k for all positive k ∈ R. This lets us evaluate rational functions that include inﬁnitesimals. For instance, if f (x) = x2 /(x − 1) we can extend f to include calculations with the inﬁnitesimal v by substitution: f (x + v) = (x + v)2 /(x + v − 1) ∈ R(v)

To extend real functions f : D → R so that we can calculate f (x + h) when x ∈ D and h is inﬁnitesimal, we work in a suitable larger ﬁeld K. The ﬁeld K = R(v) is ﬁne for rational functions, but not for more general functions. Leibniz’s version of calculus deals with polynomials, trigonometric functions, exponential functions, and, more generally, any function that can be expressed as a power series. In this case we need a larger ﬁeld that includes power series in an inﬁnitesimal. If we include a single inﬁnitesimal ε, then we also need all the elements that are produced by performing the usual operations of addition, subtraction, multiplication, and division with power series so that the system is an ordered ﬁeld. In particular, the ﬁeld should include all power series in ε and inverses of inﬁnitesimals such as 1/ε . Foundational Example 15.3. (The Superreal Field Generated by a Single Inﬁnitesimal). The superreal ﬁeld Rε is deﬁned to consist of all series of the form

∞ ± r =n

ar εr

where n is an integer (which may be negative). To obtain a ﬁeld, the series are added, subtracted and multiplied term by term. Division in the form

²∞ ±

r =m

ar ε

³´ ² ∞ ± r r=n

br ε

³

r

=

∞ ±

r=m +n

cr ε r

15.5 Inﬁnitesimals as Elements of an Ordered Field

325

is achieved by solving the equation

² ∞ ± r=m +n

cr ε

³² ∞ ± r r =n

brε r

³

=

²∞ ± r =m

arε r

³

recursively for successive coefﬁcients cm +n, cm +n+1, . . .. Here we do not need to worry about convergence, because all the operations are purely formal. It is possible to give an alternative set-theoretic deﬁnition in which the superreals are deﬁned to be functions s : Z → R, where each s has an integer n such that s(r) = 0 ∑ r for r < n. The function s : Z → R then corresponds to the inﬁnite series ∞ r =n ar ε . Knowing that the superreal numbers can be given a formal set-theoretic deﬁnition, we can safely think of them as inﬁnite series in ε.

r An element ∞ r =n a r ε is an inﬁnitesimal of order n if a n n ≥ 1. It is ﬁnite if n ≥ 0 and inﬁnite of order k if n < 0, an ² = 0 and k = −n.

D E FI N I T I O N 15.4.

²= 0 and

(i) ε3 + 17ε 25 is a third order inﬁnitesimal. π + ε, ε 4 are ﬁnite. ε−11 − 15ε are inﬁnite. = ε − ε3 /3! + · · · is a ﬁrst order inﬁnitesimal.

Examples 15.5. (ii) 0, 25, (iii) 1 and (iv) sin

/ε ε

The term ‘order’ is used here in a special sense referring to the ‘order of size’ of an inﬁnitesimal as distinct from the order a < b in the ﬁeld. There may be two elements a < b where the order of a as an inﬁnitesimal is greater than the order of b. An example is a = −ε2 and b = ε. If there is a danger of ambiguity, we refer to the order of a single element a in the sense of Deﬁnition 15.4 as the order of inﬁnitesimality of a, and the order a < b between two elements as the linear order between a, b. ∑ r The order of inﬁnitesimality of a = ∞ r =n ar ε is the subscript n of the dominant term an, which is the ﬁrst non-zero term in the sum. linear order a < b also depends on the dominant terms of a, b. If a = ∑The ∞ arεr , b = ∑∞ br εr , then without loss of generality we can assume m ≤ n. r =n r =m If m = n then the linear order of a, b is the same as the linear order of am , bn . If m < n, then am dominates and the linear order is given by the sign of am . (In the latter case, bm = 0, so the same rule applies: in this case the linear order of a, b is the same as the linear order of am , 0.)

ε < ε

2 Examples 15.6. (i) 3 . (ii) 1 0 are ﬁnite. (iii) r for every positive r

− /ε < ε
r for all r ∈ R negative inﬁnite if x < r for all r ∈ R ﬁnite if a < x < b for some a, b ∈ R positive inﬁnitesimal if 0 < x < r for all positive r ∈ R negative inﬁnitesimal if 0 < −x < r for all positive r ∈ R.

Using the standard properties of an ordered ﬁeld it is straightforward to prove that an element x is inﬁnitesimal if and only if 1/x is (positive or negative) inﬁnite. (These and other properties of arithmetic and order are left as exercises at the end of the chapter.)

15.6 Structure Theorem for any Ordered Extension Field of R

15.6

327

Structure Theorem for any Ordered Extension Field of R The key to visualising the inﬁnitesimal structure in the ﬁeld Rε is a simple but powerful structure theorem: T H E O R E M 15.9. Every element x in a super ordered ﬁeld K is either inﬁnite, or uniquely of the form x = c + h where c ∈ R and h is zero or inﬁnitesimal.

Proof. We need consider only the case where x is ﬁnite, so that a < x < b, where a, b ∈ R. Let S = {r ∈ R : r < x}. Then S is non-empty because a ∈ S, and bounded above (by b). By completeness of R, the set S has a unique least upper bound c. Let h = x − c, so x = c + h. Now we use completeness to show that if h ² = 0, then h is inﬁnitesimal. There are two cases: For h > 0, suppose that there exists k ∈ R such that 0 < k < h. Then c + k < c + h = x, so by deﬁnition of S, c + k ∈ S. But this implies that c + k is greater than the least upper bound c. By contradiction, h is inﬁnitesimal. For h < 0, suppose there exists k ∈ R such that 0 < k < −h. Then x = c + h < c − k < c, so by deﬁnition c − k is an upper bound of S. But c − k is less than the least upper bound c, again giving a contradiction. So h is inﬁnitesimal. To prove uniqueness, suppose that c + h = d + k, where c, d ∈ R and h, k are inﬁnitesimal. Then c − d = k − h is inﬁnitesimal or zero, and real. Hence it is zero. We have now proved that any ﬁnite x ∈ K is uniquely of the form x = c + h, where c ∈ R and h is inﬁnitesimal or zero. We extract a key concept: D E FI N I T I O N

c = st (x).

15.10. The real number c is called the standard part of x and is written

This structure theorem is the key to imagining inﬁnitesimal quantities on a number line. Over the centuries, we have expanded our interpretation of a number line, from the Greek idea of a line where the distance between two points is the same regardless of direction, to a number line with positive and negative numbers, rational numbers, irrational numbers, and real numbers. There is no logical reason to stop there. A super ordered ﬁeld, has ﬁnite, inﬁnite, and inﬁnitesimal elements. The structure theorem tells us that any ﬁnite element is (uniquely) a real number plus an inﬁnitesimal. An inﬁnitesimal is, of course, too tiny to see to a normal scale, but this is not as problematic as it seems to be at ﬁrst sight (pun intended). For the real numbers, on a normal scale we cannot see the difference between the irrational number π and the rational number given by π to a thousand decimal places. However, if we magniﬁed a small part of the picture by a factor 101000 , the difference would become apparent. This simple idea offers the way ahead: we can magnify a super ordered ﬁeld to distinguish inﬁnitesimally close points visually. First, we establish some simple ideas relating ﬁnite elements and inﬁnitesimals. 15.11. Let F be the subset of ﬁnite elements in K and I the subset of elements that are zero or inﬁnitesimal. Then a, b ∈ F, c, d ∈ I imply a + b, a − b, ab ∈ F, c + d, c − d, ac ∈ I. P RO P O S I T I O N

328

Inﬁnitesimals in Real and Complex Analysis

Proof. These follow from the deﬁnitions. Proposition 15.11 states that the sum, difference, and product of ﬁnite numbers are also ﬁnite, but the quotient a/b need not be ﬁnite (if b is inﬁnitesimal). The sum and difference of inﬁnitesimal elements are inﬁnitesimal, and the product of a non-zero ﬁnite element by an inﬁnitesimal is again inﬁnitesimal. (Anyone familiar with ring theory will recognise that F is a ring and I is an ideal of F; however, this is not essential to follow the ideas in this chapter.) P RO P O S I T I O N

15.12. For x, y ∈ F, the standard part map st : F

→ R satisﬁes

st (x + y)

= st (x) + st (y) st (x − y) = st (x) − st (y) st (xy) = st (x) st (y) st (x/y) = st (x) /st (y) (for st (y) ² = 0)

Proof. Use algebraic manipulation of the deﬁnition, taking care with division.

15.7

Visualising Inﬁnitesimals as Points on a Number Line The structure theorem offers new ways of visualising inﬁnitesimal quantities as points on a number line. A standard picture cannot distinguish points that differ by an inﬁnitesimal, because the difference is too small to see, or represent inﬁnite quantities, because they are too far away. But a linear map m(x) = ax + b for appropriate values of a, b ∈ K reveals inﬁnitesimal or inﬁnite detail: 15.13. The map m : K → K where m(x) = (x − c)/d for c, d ² = 0 is the d-lens pointed at c. If d is inﬁnitesimal, then m is a microscope pointed at d. If c is inﬁnite then m is a telescope pointed at inﬁnity. The set

D E FI N I T I O N

V

= {x ∈ K : m(x) is ﬁnite}

is the ﬁeld of view of m. The ﬁeld of view V of a d-lens contains those elements x ∈ K whose image m(x) is ﬁnite, and these images can be marked on a ﬁnite line in a picture. For instance, if c is a real number and δ is inﬁnitesimal, the ﬁeld of view V of a δ -microscope pointed at c contains c, c + δ , c + 2δ . These differ by inﬁnitesimal quantities, but they map to m(c) = 0, m(c + δ ) = 1, m(c + 2δ ) = 2, which are now visibly different. At the same time, the points c +δ and c +δ+ 3δ 2 map to m(c +δ ) = 1, m(c+δ+ 3δ 2 ) = 1 + 3δ , which differ by an inﬁnitesimal and cannot be distinguished at this scale. See Figure 15.11. When we make a map in a real-world situation, say a map of England including London, we mark the location of London using the same name. This technique can be used to advantage when magnifying inﬁnitesimal detail using a δ -lens. Renaming the image points with their original names, as in Figure 15.12, the picture may be interpreted

15.7 Visualising Inﬁnitesimals as Points on a Number Line

329

δ

Figure 15.11 Scaling up K using a -lens pointed at c.

Figure 15.12 Scaling up inﬁnitesimal detail.

as scaling up part of the number line K to distinguish points that differ by an inﬁnitesimal amount. Using a speciﬁc map, it is possible to see only a certain level of detail, because some differences in the image are too small to distinguish, and some are too far away. Leibniz imagined inﬁnitesimals of order 1, 2, 3, . . . and in the above pictures we have included only examples of polynomials in an inﬁnitesimal that follow Leibniz’s intuition. However, we must be prepared for more general situations that occur in practice. For example, if h is an inﬁnitesimal of order 1 and h has a square root r ∈ K , then r2 = h and r must have order 21 . To deal with the general case, instead of referring to the speciﬁc order of inﬁnitesimality as a whole number, we compare the relative order of size of two elements a, b as follows: D E FI N I T I O N 15.14.

If b ∈ K and b ² = 0, an element a ∈ K is:

of higher order than b if a/b is inﬁnitesimal, of the same order as b if a/ b is ﬁnite, of lower order than b if a/b is inﬁnite. When visualising inﬁnitesimal detail using a δ -lens pointed at c, the ﬁeld of view V consists of elements that differ from c by an element of the same or higher order. Those differing by the same order map onto distinct points: those that differ by a higher order quantity are indistinguishable to the naked eye. We can model this visually using: 15.15. The optical δ -lens or optical microscope µ pointed at c is µ : → R where µ(x) = st ((x − c)/δ ).

D E FI N I T I O N

V

330

Inﬁnitesimals in Real and Complex Analysis

Figure 15.13 Scaling points in the ﬁeld of view onto the whole real line.

The image is now a drawing of the real line. Points in V that differ by an element of the same order as δ map on to different real numbers and points that differ by a higher order than δ map onto the same real number. Meanwhile, the δ -lens m(x) = (x − c)/δ maps points outside the ﬁeld of view V onto inﬁnite elements of K, which are beyond the range of a ﬁnite drawing. In Figure 15.13 the line representing the ﬁeld K is greyed out to show that only the point c and the ﬁeld of view V are magniﬁed to give a visible point on the real line. In general, the optical microscope µ : V → R always maps the ﬁeld of view V onto the whole of R because µ(c + xδ ) = x for any x ∈ R. A special case occurs when we take c = 0 and δ = 1. Now the ﬁeld of view is the set of ﬁnite numbers F, and the optical δ -lens is the standard part function st. The image is the whole real line and the points mapping onto a particular real number x ∈ R consist precisely of x and those points differing from x by an inﬁnitesimal. This suggests: D E FI N I T I O N

15.16. The monad Mx around x ∈ K is Mx

= {x + h ∈ K : h ∈ I}

where I is the subset of K consisting of zero and inﬁnitesimals. Some sources use the alternative name ‘halo’ in place of ‘monad’. In this deﬁnition, x may be any element of K, so any inﬁnite value of x is also surrounded by its own monad. If x is ﬁnite, then by the structure theorem of Theorem 15.9, there is only one real number c ∈ Mx and the standard part map st : F → R collapses the monad Mc down onto the real number c. D E FI N I T I O N

x − y ∈ I.

15.17. The relationship x

≈ y is deﬁned on K by x ≈ y if and only if

This is an equivalence relation and the monads are the equivalence classes. It lets us imagine the picture of the super ordered ﬁeld K as points on a line, where the ﬁnite points F consist of the real numbers and each real number c is surrounded by the monad Mc of elements c + h where h is inﬁnitesimal. This vision lets us see that any super ordered ﬁeld K is not complete. We need: 15.18. For a ∈ R, the monad Ma is bounded above by any real number b > a, but has no least upper bound. THEOREM

Proof. If l ∈ K is a least upper bound of Ma, then either l ∈ Ma or l ∈ Mb for b > a. In the ﬁrst case, for any positive inﬁnitesimal h, l + h ∈ Ma is greater than the supposed

15.8 Complex Inﬁnitesimals

331

upper bound l of Ma . In the second, l − h ∈ Mb is an upper bound for Ma that is smaller than the least upper bound l. By contradiction, Ma has no least upper bound. C O RO L L A RY 15.19.

A super ordered ﬁeld is not complete.

This should not be a surprise. A complete ordered ﬁeld is unique up to isomorphism and does not contain inﬁnitesimals. So a super ordered ﬁeld that contains inﬁnitesimals cannot be complete. The structure theorem, Theorem 15.9, tells us why. Nevertheless, we now have a fundamentally new way of visualising ‘the number line’. Not only does it include the familiar real numbers; it can include inﬁnitesimals and inﬁnite quantities that can be seen by human eyes only through appropriate d-lenses.

15.8

Complex Inﬁnitesimals The generalisation of inﬁnitesimals in any super real ﬁeld K can be extended to the complex case by using quantities of the form x + iy where x, y ∈ K and i2 = −1. D E FI N I T I O N 15.20. Given a super ordered ﬁeld K, the corresponding super complex ﬁeld is the ﬁeld K[i] of polynomials in i where i 2 = −1. It is an extension ﬁeld of the ﬁeld of complex numbers C = R[i]. It is a ﬁeld because the polynomial t 2 + 1 has no zeros in any ordered ﬁeld, so is irreducible. Alternatively, a direct proof follows the usual one for C by writing (x + iy)−1 in the form u + iv. An element z = x + iy where x, y ∈ K is inﬁnitesimal if both x and y are inﬁnitesimal, ﬁnite if both x and y are ﬁnite, and inﬁnite if at least one of x, y is inﬁnite. The standard part of a ﬁnite element z = x + iy is

st (z) = st (x) + i st (y)

An element w is of higher order than a non-zero element z if w/z is inﬁnitesimal; of the same order if w/z is ﬁnite and non-zero; and of lower order if w/z is inﬁnite. Let Fi be the set of ﬁnite elements in K[i] and Ii the set of elements that are inﬁnitesimal or zero. (Again F i is a ring and Ii is an ideal in F i with the properties speciﬁed in Proposition 15.11.) 15.21 (Structure Theorem for a Super Complex Field). Every element z = x + iy in a super complex ﬁeld is either inﬁnite or uniquely of the form z = w + h where w ∈ C is given by w = st (z) and h is zero or inﬁnitesimal. For ﬁnite u, v, the standard part map satisﬁes THEOREM

st (u + v)

= st (u) + st (v) st (u − v) = st (u) − st (v) st (uv) = st (u) st (v) st (u/ v) = st (u)/st (v) (for st (v) ² = 0) st (x + iy) = st (x) + i st (y) Proof. If z is ﬁnite, then both x and y are ﬁnite elements in K, and the structure theorem for super ordered ﬁelds shows that they are uniquely of the form x = c + p, y = d + q

332

Inﬁnitesimals in Real and Complex Analysis

where c, d ∈ R and p, q ∈ I, so z = w + h where w = c + id The properties of the map st follow by direct calculation.

∈ C, and h = p + iq ∈ Ii .

Many deﬁnitions and properties generalise from the real case, such as: D E FI N I T I O N

15.22. The monad Mz around z ∈ K[i] is

{z + h ∈ K[i] : h ∈ Ii } The monads partition K[i] into non-overlapping subsets consisting of points that differ by an inﬁnitesimal. This includes the case where z may be inﬁnite, so that even an inﬁnite element is surrounded by a monad of points differing from it by an inﬁnitesimal. Finite elements in K[i] consist of the underlying ﬁeld of complex numbers together with the monad Mz surrounding each complex number z. D E FI N I T I O N

15.23. If d ² = 0, the d-lens pointed at w in a super complex ﬁeld is m(z) =

z− w d

The ﬁeld of view of m is the set of points z in the super complex ﬁeld such that (z − w)/ d is ﬁnite. The optical d-lens pointed at w is

µ(z) = st (m(z) = st

·

z− w d

¸

If d is inﬁnitesimal, the d-lens is called a microscope, and if w is inﬁnite, it is called a telescope. Multiplying by a complex number in the form rei θ scales by a factor r and turns through an angle θ , so it is sensible to take θ = 0 and to select d to be of the form d = δ + i0 where δ > 0. (Such an element will be called a positive inﬁnitesimal.) Then the δ -lens simply scales by a factor δ and does not rotate the image. For this reason, we will always assume that the scaling factor δ is a positive element of the subﬁeld K.

=

θ∈

Example 15.24. Figure 15.14 shows the point z0 reiθ , where r, R, lying on the circle z r in K[i], z0 h reiθ+ε lies on the circle where is an inﬁnitesimal in K. Let be an optical microscope (z) st (z z0 ) , where is the same order as . Then

µ

| |=

+ =

µ =

h = rei θ+ε

− /δ

δ

ε

ε

− reiθ = rei θ (ei ε − 1) = z0 (iε − ε2 /2 + · · · ) = z0 iε + higher order terms So µ (z0) + h = µ (z0 + z0iε ), and the point z0 + h on the circle is mapped optically to the same point as z0 + z0 iε.

15.9 Non-standard Analysis and Hyperreals

333

Figure 15.14 An inﬁnitesimal part of a circle under optical magniﬁcation.

Multiplying by i ε rotates the plane through a right angle (multiplying by i) and scales by the real factor ε , so z0 + z0 iε is on the tangent at right angles to the radius from 0 to z0 . Furthermore, for any λ ∈ R, taking ε = λδ maps z0 + z0iε onto a general point z0 + z0 iλ on the tangent, so that the optical magniﬁcation transforms the part of the unit circle in the ﬁeld of view of µ onto the whole tangent line. In this very precise sense, an inﬁnitesimal portion of the graph of a circle is ‘locally straight’ and is indistinguishable from an inﬁnitesimal part of the tangent line when viewed through an optical microscope.

15.9

Non-standard Analysis and Hyperreals In the early days of calculus, functions were given by formulas. When the formula is a polynomial or a power series – which includes functions such as exponentials and trigonometric functions – the value of f (x + h) for inﬁnitesimal h can be calculated in the superreal numbers Rε in the real case, or the supercomplex numbers Cε in the complex case. The derivative may then be calculated as

·

f (x + h) − f (x) f ±(x) = st h

¸

for inﬁnitesimal h

The ﬁelds Rε and Cε are sufﬁcient for differentiation. By the Fundamental Theorem of Calculus, they are also sufﬁcient for integration, using an antiderivative. However, there is no mechanism in these ﬁelds to deal with more general functions such as sequences s : N → R, and they do not permit a suitable deﬁnition of the Riemann integral. When the real numbers were formulated set-theoretically at the end of the nineteenth century, inﬁnitesimals had no place in formal epsilon-delta analysis. In 1966, Abraham Robinson [17] made the formal use of inﬁnitesimals rigorous using mathematical logic. He introduced an extension ﬁeld ∗R of R, the hyperreal numbers, which includes inﬁnitesimals and provides natural extensions of all of the basic functions of analysis. It is based on a distinction between two different types of logical statement, as follows. The axioms for arithmetic and order in R quantify properties of elements of R. These include for all x, y ∈ R : x + y = y + x

334

Inﬁnitesimals in Real and Complex Analysis

or or

there exists 0 ∈ R such that: for all x ∈ R, x + 0 = x for all x, y, z ∈ R : x > y, y > z implies x > z

Just one axiom – the axiom of completeness – quantiﬁes properties of sets: for all sets S ⊆ R, if S is non-empty and bounded above, then it has a least upper bound. Corollary 15.19 proves that super ordered ﬁelds are not complete, because monads are bounded above but fail to have least upper bounds. This is a subtle clue. When Robinson introduced the hyperreal numbers, he observed that properties that quantify elements ( ﬁrst order logic) generalise to ∗ R, but properties quantifying sets (second order logic) may not. Writing properties using the universal quantiﬁer ∀ (meaning ‘for all’) and ∃ (‘there exists’), most properties of arithmetic quantify elements, such as

∀x, y, ∈ R : x + y = y + x or

∃0 ∈ R : ∀x ∈ R : x + 0 = x

These use ﬁrst order logic, but the completeness axiom – which quantiﬁes sets – does not. The solution is to ﬁnd a system of hyperreal numbers ∗ R for which any ﬁrst order logical statement that is true for the real numbers R remains true in the extended system ∗ R. For example

∀x, y, ∈ ∗ R : x + y = y + x

or

∃0 ∈ ∗ R : ∀x ∈ ∗ R : x + 0 = x.

remain true. However, second order logical statements, such as the completeness axiom, need not extend to ∗ R. 15.25. The system of hyperreal numbers ∗ R is a proper ordered ﬁeld extension of R where every subset D ⊆ R has an extension D ⊆ ∗ D ⊆ ∗ R and every function f : D → R has an extension f : ∗ D → R that satisﬁes: Transfer Princip le: Every true statement about the real numbers R expressed in ﬁrst order logic is true in the extended system ∗ R.

D E FI N I T I O N

(It is convenient use the same symbol for the extended function as for f to avoid proliferation of ∗s. This should not cause confusion.) We talk of ‘the’ hyperreals, but the construction does not lead to a unique result; however, any of them does the same job, so we assume that a particular one has been selected.

15.9 Non-standard Analysis and Hyperreals

335

There are two possible approaches to the hyperreal numbers. The ﬁrst is to assume that such a system exists and then build up a theory based on the deﬁnition. The second is to provide a formal existence proof, or better, a construction, of a hyperreal number system. The ﬁrst approach, though it does not prove the actual existence of hyperreal numbers, has great potential. For example, assuming its existence, it is possible to prove that the extension ∗ N of the natural numbers N must have inﬁnite elements. The proof is simple. The ﬁrst order statement

∀x ∈ R ∃n ∈ N : n > x

states that for every real number x, there is always a bigger natural number n. Its extension says that

∀x ∈ ∗ R ∃n ∈ ∗N : n > x

and, since we know that ∗ R contains inﬁnite elements, so does ∗ N. This immediately gives a new way of thinking about the limit of a sequence as a function s : N → R that extends to s : ∗ N → ∗ R . Consider the elements s(N) for inﬁnite N and take the standard part st (s(N)). If this is the same for all inﬁnite N, then this common value is the limit of the sequence. For instance, if 6n2 + n sn = s(n) = 2 3n + 1 then the limit is st s(N)

= st

·

6N 2 + N 3N 2 + 1

¸

= st

·

6 + 1/N 3 + 1/N 2

¸

= 63 ++ 00 = 2

Limits still have to be calculated, but the deﬁnitions become simpler. For instance, the usual deﬁnition of the limit s of a sequence sn of real numbers is

∀ε > 0 ∃N ∈ N : n > N ⇒ |sn − s| < ε

Using the hyperreals, this becomes, as just explained:

For all inﬁnite n ∈ ∗ N : s = st (sn )

In general, deﬁnitions using the hyperreals (called non-standard analysis) involve fewer quantiﬁers than the standard deﬁnition. For instance, pointwise continuity of f : D → R has a standard deﬁnition:

∀x ∈ D ∀ε ∈ R (ε > 0) ∃δ ∈ R (δ > 0) ∀x± ∈ D : |x − x± | < δ ⇒ |f (x) − f (x± )| < ε and uniform continuity of f : D → R has a standard deﬁnition: ∀x ∈ D ∀x± ∈ D ∀ε ∈ R (ε > 0) ∃δ ∈ R (δ > 0) : |x − x± | < δ ⇒ |f (X) − f (x±) | < ε

Each has four quantiﬁers that are highly complicated for the human mind to hold and manipulate all at once. The corresponding non-standard deﬁnitions are, for pointwise continuity:

∀x ∈ D ∀x± ∈ ∗D : x − x± inﬁnitesimal ⇒ f (x) − f (x± ) inﬁnitesimal

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Inﬁnitesimals in Real and Complex Analysis

and for uniform continuity:

∀x, x± ∈ ∗ D : x − x± inﬁnitesimal ⇒ f (x) − f (x± ) inﬁnitesimal which have only two quantiﬁers and are much simpler.

15.10

Outline of the Construction of Hyperreal Numbers It is possible to use the idea of hyperreals based on the transfer principle, without proving that hyperreals exist. But to put the idea on a sound logical basis, it is important to show that such a system can be constructed from known mathematics. Broadly speaking, the construction of ∗ R from R follows a similar overall strategy to the one that Cantor used to construct the real numbers R from the rational numbers Q. His construction begins with the set of all Cauchy sequences (an ) of rational numbers; where a sequence is Cauchy if for any ε > 0 there exists N ∈ N such that m, n > N ⇒ |am − an | < ε. Then he sets up an equivalence relation: (an ) ∼ (bn)

⇐⇒

(an − bn)

→0

Then R is deﬁned to be the set of equivalence classes of Cauchy sequences, and the operations of arithmetic and order are derived from operations on Cauchy sequences. The construction of the hyperreal numbers ∗R from R follows a similar overall pattern. It begins with all sequences (rn ) of real numbers, sets up an appropriate (and cunningly deﬁned) equivalence relation ∼, and deﬁnes ∗ R to be the set of equivalence classes. The ﬁeld ∗ R is considered as an extension of R by identifying the real number r with the equivalence class [rn ], where rn = r for all n. The detailed construction requires the notion of an ‘ultraﬁlter’, akin to the axiom of choice. For a more detailed discussion, see Robinson [17] or Stewart and Tall [21]. Once the equivalence relation is set up, extending real sets and functions to hyperreal sets and functions is exquisitely simple. The extension of any subset S ⊆ R to ∗S ⊆ ∗ R is deﬁned by:

∗S = {[rn ] ∈ ∗ R : rn ∈ S for all n ∈ N}

The extension set ∗S is precisely the set of equivalence classes of sequences whose terms all lie in S. The extension of any function f : S → R to f : ∗ S → ∗ R is equally straightforward. Just deﬁne f ([rn ]) = [f (rn)] for all [rn]

∈ ∗S

If subsets of the real numbers or real functions are formulated set-theoretically using statements that quantify only elements, then those sets and functions can be extended to the hyperreal numbers using the same statements. For instance, if S = { x ∈ R : x ≥ 0} √ and f (x) = x, then ∗ S = {x ∈ ∗ R : x ≥ 0}. Because every x ∈ S satisﬁes x ≥ 0 and f (x)2 = x, it follows that for every x ∈ ∗ S, the same equation holds, so f (x) is the √ (positive) square root of x. For a positive inﬁnitesimal x, this means that x is also a positive inﬁnitesimal whose square is x.

15.11 Hypercomplex Numbers

337

The theory also extends to more general real functions f : S → Rn , where S ⊆ Rm , to give an extension f : ∗S → ∗ Rn that satisﬁes the transfer theorem for logical statements quantifying several elements in R. In particular, since every complex function w = f (z) can be expressed as a real function of two variables taking (x, y) to (u, v), where z = x + iy and w = u + iv, the hyperreal theory generalises naturally to the complex case.

15.11

Hypercomplex Numbers To treat complex analysis in a similar manner we need the complex analogue of ∗ R. The hypercomplex numbers ∗ C are of the form a + ib, where a, b ∈ ∗ R and i2 = −1. (As for hyperreals, the word ‘the’ assumes a particular choice for ∗ R, hence for ∗ C.) They generalise the real case so that any function f : S → C extends to a function f : ∗ S → ∗C that satisﬁes the transfer principle: any logical statement quantifying elements in C remains true in ∗ C. For example, suppose that a complex analytic function ∑ n f : D → C on a domain D is expressed as a power series f (z0 + h) = ∞ n=0 a nh for ∗ ∗ h ∈ C, |h| < R. Then the extended function f : D → C is given by the same formula for h ∈ ∗ C, | h| < R. This makes the inﬁnitesimal extension of complex analysis simpler than the real case. In particular, for any inﬁnitesimal ε ∈ C, the values of functions can be calculated using power series in ε, so we need only work in the ﬁeld Cε . This, of course, is a subﬁeld of the hypercomplex numbers ∗ C, deﬁned in the same way as Rε but using complex coefﬁcients in the power series in ε . For any inﬁnitesimal δ , the optical δ -lens µ pointing at z0 ∈ C is deﬁned by

µ(z) = st z −δ z0

Its ﬁeld of view is the set of elements z0 + h where h/δ is ﬁnite, so h = δ k where k is ﬁnite. This lies within the monad Mz0 of points lying an inﬁnitesimal distance from z0 . The ﬁeld of view need not be the whole of the monad. For instance, if we take any inﬁnitesimal ε and choose δ = ε2 then z0 + ε does not lie in the ﬁeld of view of the optical δ -lens µ because ε/δ = 1/ε is inﬁnite. Essentially the optical microscope lets us see detail differing from z0 by the same order as δ , but lower order detail is too small and higher order detail is too far away to be seen in an optical image. Writing 1 /δ in polar coordinates as 1/δ = rei θ , the lens magniﬁes distances from z0 by the (inﬁnite) factor r and rotates through the angle θ . Choosing δ to be any positive inﬁnitesimal in ∗ R gives θ = 0 so that the optical microscope µ magniﬁes the picture by a factor r without any rotation. For this reason, when using an optical microscope we take the magniﬁcation factor δ to be a positive inﬁnitesimal in ∗R. A subset S ⊆ ∗C is called an inﬁnitesimal δ -shape near z 0 it is of the form S = {z0 + h : z0 ∈ C, h is a δ -inﬁnitesimal}. D E FI N I T I O N 15.26.

∈ C if

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Inﬁnitesimals in Real and Complex Analysis

δ

δ

Figure 15.15 Magnifying an inﬁnitesimal -shape using an optical -microscope.

δ ∈ ∗ R, the subset {z0 + h ∈ ∗ C : z0 ∈ C, −δ ≤ h ≤ δ} is an inﬁnitesimal δ -shaped square in ∗ C of side 2δ , centred on z0 . Using the standard convention that the image µ(z) is also denoted by the symbol z, this lets us see the inﬁnitesimal shape in the complex plane C using the optical microscope µ , see Fig-

Example 15.27. For an inﬁnitesimal

ure 15.15.

An analytic function f : D → C then extends naturally to f : ∗D → ∗ C, and an inﬁnitesimal shape near z0 in ∗ D transforms to an inﬁnitesimal shape in ∗ C near w0 = f (z0). In the z-plane, let z0 + h be in the ﬁeld of view of the optical microscope µ (z) = st ((z − z0 )/δ). Then

µ(z0 + h) = st (h/δ) Using an optical microscope ν (w) = st ((w − w0)/δ ) in the w-plane with the same magnifying factor δ , ν ( f (z0 + h)) = st (( f (z0 + h) − f (z0 ))/δ) = st ((( f (z0 + h) − f (z0 ))/h)st (h/δ) = f ± (z0)µ(z0 + h) The complex number µ (z0 + h) is transformed to its optical image ν ( f (z0 + h)) multiplied by f ± (z0). Writing f ± (z0 ) in polar coordinates, f ± (z0) = reiθ , this scales the picture by a factor r = |f ± (z0)| and turns it through an angle θ = arc ( f ± (z0)), see Figure 15.16. If f ±(z0 ) ² = 0, the transformation of the optical image is conformal, turning the inﬁnitesimal shape precisely through the angle arg f ± (z0) and scaling lengths by the factor | f ± (z0) |. If f ± (z0) = 0, the shape collapses to a single point. In this case, the detail in the w-plane involves inﬁnitesimals of higher order. In Section 15.2 the Taylor series of f near z0 is written in the form

15.11 Hypercomplex Numbers

339

Figure 15.16 Magnifying a transformed inﬁnitesimal shape.

f (z0 + h) = f (z0) + hnf (n) (z0 )/n! + · · ·

where n ≥ 1 and f (n)(z0 ) ² = 0. This is interpreted as the sum of the constant f (z0 ) plus the inﬁnitesimal hn f (n)(z0 )/n! + · · · of order n in the variable h. Using an optical microscope of order δ n gives

ν (w) = st ((w − w0 )/δn )

so

ν ( f (z0 + h)) = st (( f (z0 + h) − f (z0)/δ n) = st (hn f (n) (z0))/n!/δ n) = st (hn /δn )f (n) (z0))/n! = st (h/δ )nf (n) (z0 )/n! = µ(z0 + h)n f (n) (z0)/n! Write µ(z0 + h) = reiθ and f (n) (z0)/ n! = kei α in polar coordinates, to get ν ( f (z0 + reiθ )) = rn einθ keiα so and

|ν ( f (z0 + rei θ ))| = krn arc ν ( f (z0 + h))

= nθ + α

This effect can be seen by starting with an inﬁnitesimal δ -shape in the z-plane in the form of a sector of a circle centre z0, with two inﬁnitesimal sides of radius δ , at (ﬁnite) angles θ and θ + φ to the real axis. The shape is then transformed in the w-plane as an inﬁnitesimal sector that can be seen through an optical δ n -lens as a sector of a circle, centre w0 = f (z0 ), with the radius to f (z0 + h) turned through an angle nθ +α, the angle between the two radii stretched to nφ, and the lengths of each radius scaled from r to krn , see Figure 15.17. In the case of an inﬁnitesimal δ -shape in the form of a circular path f (γ (t)) (t ∈ [0, 2π ]) then, as the circle is traced once round the critical point, the image path f γ

340

Inﬁnitesimals in Real and Complex Analysis

Figure 15.17 Transformation of an inﬁnitesimal shape at a critical point of order n.

Figure 15.18 Optical image of a point moving round an inﬁnitesimal circular path.

traces round the corresponding inﬁnitesimal circle in the w-plane n times. This relates to the theory of Riemann surfaces, where a point z in the z-plane moves once round a critical point as its image w = f γ (z) traces round the corresponding path in the w-plane n times. It is visualised in Figure 15.18 with part of the z-plane magniﬁed by an optical δ-microscope pointed at z0 and the corresponding part of the w-plane magniﬁed by an optical δ n-microscope pointed at w0 . In the ﬁgure the image in the w-plane is represented by a horizontal plane in three dimensions and the path on the Riemann surface directly above has n levels. As γ (t) moves in the z-plane from 0 to 2π/n the corresponding

15.12 The Evolution of Meaning in Real and Complex Analysis

341

point on the Riemann surface moves once round the initial level and then moves up on successive levels as t moves from 2r π/n to 2(r + 1)π/n, returning to the initial level when t = 2π . This phenomenon applies to every point z0 ∈ S. Furthermore, each inﬁnitesimal δ circle lies within the monad Mz 0 , and each monad contains just one complex number z0 . If z ² = z0 then an inﬁnitesimal δ -circle around z lies in the distinct monad Mz . The behaviour of the function near any point z ∈ C therefore depends only on the order of inﬁnitesimality of the function f at the point z. This completes the picture of the behaviour of regular analytic functions from an inﬁnitesimal viewpoint. An analytic function f : D → C has a natural extension f : ∗ D → ∗ C which generalises to a function from ∗ D to the Riemann surface of f with multiple levels involving points in ∗ C . This extension then has a unique analytic inverse from the Riemann surface back to ∗D.

15.12

The Evolution of Meaning in Real and Complex Analysis At this point we are in a position to look back on the historical evolution of real and complex analysis to gain a broader view of the current state of the theory as part of an ongoing development that will continue to evolve in the future. The historical path taken depends on the prevailing ideas at the time and may later be seen in an entirely different light. We review relevant parts of Chapter 0 and then move on to the modern era.

15.12.1

A Brief History Initially the calculus developed by Leibniz and Newton focused on the relationship between variables represented as curves in the plane. On the continent, the inﬁnitesimal methods of Leibniz prevailed, while in England, Newton began by thinking of a variable x changing in time, which he called a ‘ﬂuxion’, and its rate of change x± , which he called a ‘ﬂuent’. He also formulated the Binomial Theorem in the form (x + o)n

= xn + nxn−1 o + 21 n(n − 1)xn−2o2 + · · ·

(x + o)n − xn o

= nxn−1 + o · 12 n(n − 1)xn−2 + · · ·

This holds good not only for a whole number n, but also for fractional and negative powers, which expresses (x + o)n as a power series in o, and as o becomes small, leads to the term in o being small enough to neglect, giving the derivative of xn as nx n−1 . In the eighteenth century, work on the calculus by Euler and others focused on real and complex power series manipulated as symbols to give signiﬁcant results whose logical status would be questioned by later generations. It was not until the turn of the nineteenth century that Wessel, Argand, Gauss, and others began interpreting the complex numbers as points in the plane. Derivatives were calculated using the same formula as the real case, while complex integrals were taken along contours in the plane from z0 to z1 . As in

342

Inﬁnitesimals in Real and Complex Analysis

¹

the real case, if f has an antiderivative F in its domain D then zz01 f (z)dz = F(z1 ) − F(z0 ), so the Fundamental Theorem of Calculus generalises directly. But then the theory of real and complex functions began to diverge spectacularly. In real analysis, functions with unexpected properties were found: functions differentiable once but not twice; functions differentiable n times but not n + 1; and inﬁnitely differentiable functions (such as 2 f (x) = e−1/x for x ² = 0 and f (0) = 0) that do not equal their Taylor series expansion, even though it converges. There are even functions that are continuous everywhere and differentiable nowhere. Mathematicians realised that precise deﬁnitions and proofs were needed to cope with all these possibilities. Meanwhile, it was discovered that a complex function f differentiable in a domain D can be represented as a power series in a small disc around any point in D. At the same time, the arithmetisation of analysis led to the set-theoretic deﬁnition of the real numbers as a complete ordered ﬁeld, which cannot include inﬁnitesimals. This resulted in two parallel developments in the twentieth century with very different philosophies. Pure mathematics excluded inﬁnitesimals because they did not ﬁt into the real number system, while applied mathematicians used them informally but productively to model dynamic situations. Robinson’s theory of non-standard analysis redeemed inﬁnitesimals using mathematical logic. However, it was ignored by most applied mathematicians, who had no interest in that level of rigour, thought in terms of approximations, and were happy to discard any sufﬁciently small term if that proved fruitful, without asking further questions. It was actively resisted by many pure mathematicians, who had invested considerable effort in epsilon-delta analysis and preferred to remain with a familiar theory, or who preferred to avoid the subtle logical machinery of ultraﬁlters, with its reliance on the axiom of choice. In principle, any theorem in standard real analysis that can be proved using nonstandard analysis has a standard proof, so non-standard analysis may seem to offer no advantages. However, proofs using non-standard analysis are often simpler than epsilon-delta, once the initial investment of mastering the set-up has been made. Some important theorems currently have only non-standard proofs, because an epsilon-delta proof gets bogged down in too many ε ’s. So non-standard analysis has become an accepted technical tool in mainstream research. However, it is still a very specialised area.

15.12.2

Non-standard Analysis in Mathematics Education The intuitive appeal of inﬁnitesimals, as opposed to the difﬁcult machinery of epsilondelta methods, offers some educational advantages for students meeting analysis for the ﬁrst time. Some programmes have used this approach successfully, but they have not been widely adopted. The developments in this chapter reveal these apparently opposing views to be two sides of the same underlying coin. Applied mathematics focuses on variables that can ‘become small’ while pure mathematics focuses on ﬁxed mathematical objects deﬁned in epsilon-delta terms. Both can now be seen in a broader set-theoretical context in which real and complex numbers lie in larger structures that contain inﬁnitesimals. The elements of such extension ﬁelds of R and C may be termed ‘quantities’. A quantity

15.12 The Evolution of Meaning in Real and Complex Analysis

343

x is deﬁned to be inﬁnite if | x| > r for all r ∈ R. It is ﬁnite if it is not inﬁnite, and inﬁnitesimal if 1/x is inﬁnite. Elements in R are called ‘real constants’ and those in C are ‘complex constants’. This gives a formal language to describe an extension of the number line or complex plane that resonates with the use of the term ‘inﬁnitesimal’ in historical development. The structure theorems prove that every ﬁnite quantity x is either a constant or uniquely of the form x = a + ε , where a is either a real or complex constant and ε is a (real or complex) inﬁnitesimal. The unique constant a is called the standard part of x. Optical microscopes let us see inﬁnitesimal detail as a genuine picture on a line or in the plane. This returns us to an intuitive human way of visualising inﬁnitesimal detail that resonates with earlier conceptions of the calculus that were rejected in mathematical analysis at the beginning of the twentieth century. In 1972 Stroyan [22] introduced the idea of inﬁnitesimal microscopes and telescopes. These were used in Keisler’s undergraduate text Foundations of Inﬁnitesimal Calculus based on Robinson’s non-standard analysis, Keisler [10]. In 1980 Tall [23] modiﬁed Stroyan’s deﬁnition to include optical lenses and applied the theory to the simpler system of superreal numbers. In his book Visual Complex Analysis, Needham [14] sees a complex analytic function w = f (z) transforming the z-plane into the w-plane. Figure 15.19 shows f (z) = z2 transforming a shape in the z-plane to its image in the w-plane. The black T-shape in the z-plane is transformed visually into a T-shape in the w-plane that is scaled in size and rotated through an angle. The scaling and rotation become more precise as the shape becomes smaller. In particular, near a point z where f ±(z) ² = 0, a small shape is scaled approximately by a factor | f ± (z)| and turned through the angle arg f ± (z). Needham introduced the term ‘amplitwist’ to indicate that the transformation ‘ampliﬁes’ the scale by a factor |f ± (z)| and ‘twists’ the shape through the angle arg f ± (z). He used the term ‘inﬁnitesimal’ in a technical way that ‘does not refer to some mystical, inﬁnitely small quantity’. Instead (in Needham [14], pages 20–21) he suggested ‘two intuitive ways of speaking’, formulated as follows:

. . . if two quantities X and Y depend on a third quantity δ, then lim δ→0 YX = 1 ⇐⇒ ‘X = Y for inﬁnitesimal δ ’ ⇐⇒ X and Y are ultimately equal as δ tends to zero.

Figure 15.19 A visual transformation from the z-plane to the w-plane.

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Inﬁnitesimals in Real and Complex Analysis

θ

θ

Figure 15.20 Polar coordinates r, , with increments dr, d .

This does not explicitly say what an ‘inﬁnitesimal’ is. The approach in the current chapter deﬁnes an inﬁnitesimal formally as an element in an ordered ﬁeld extension K of the real numbers or in the corresponding complex extension ﬁeld K[i].

15.12.3

Human Visual Senses Thinking of an inﬁnitesimal as a variable that is ‘becoming small’ is a natural human way to visualise inﬁnitesimal ideas. Our eyes and brains have been wired by evolution to detect moving objects. This way of thinking also has practical advantages over thinking of an inﬁnitesimal only as a tiny physical quantity. For example, using polar coordinates, Figure 15.20 shows (r, θ ) being given a further increment dr in the radius and dθ in the angle. If r, θ , dr, dθ are all ﬁnite quantities, we can draw this as a ﬁnite picture. But if r and θ are ﬁnite and dr, dθ are inﬁnitesimal, we cannot draw the whole picture to the same scale. Using an optical δ -lens to magnify the inﬁnitesimal shape sides dr by rdθ to give a ﬁnite picture requires dr, dθ , and δ all to have the same order. If r ² = 0, then r has higher order than δ , so if a δ -lens is used magnify the shape dr to reveal a ﬁnite picture, the origin, distance r away, lies outside the ﬁeld of view. On the other hand, if r is drawn to a ﬁnite scale, the optical picture of the inﬁnitesimal shape reduces to a single point. This means that while we may imagine inﬁnitesimals as arbitrarily small ﬁxed elements in an extension ﬁeld of the real or complex numbers, when making a physical drawing, it is more natural to think of them as variable quantities that become very small. Recent research into how the human eye works shows that imagining points moving smoothly is a natural human facility. When we read text, our eye has a sharp focus which only reads a few consecutive letters (around four or ﬁve with standard size type) and our eye jumps along the line taking in information, attempting to build up the meaning of successive stretches of text. If you look at a paragraph on this page and focus on how you read it, you can sense these jumps (called saccades) as the eye moves along the text. (Try it now.) However, if you look at an object moving smoothly, there is an initial jump to lock on to the object in the sharp central vision, and the eye then follows the object in a smooth movement. This means that it is natural to imagine a point moving smoothly along a line.

15.12 The Evolution of Meaning in Real and Complex Analysis

15.12.4

345

Computer Graphics New technologies are evolving which let users dynamically control movement on a highresolution visual display, smoothly zooming in on graphs or moving objects around. Software such as Mathematica and GeoGebra let a human operator code mathematical ideas symbolically, and manipulate pictures dynamically to explore mathematical relationships. Placing the w-plane above the z-plane lets a three-dimensional view be drawn on a two-dimensional dynamic display, where z = x + iy is represented on the z-plane as (x, y, 0) and w = u + iv is represented on the w-plane as (u, v, 1). As z = x + iy moves around in the z-plane, the corresponding value w = f (z) = u + iv moves around in the w-plane. The transformation from z to w can be signiﬁed by an arrow linking z to f (z). As z varies, this gives a mapping diagram from the z-plane to the w-plane as shown earlier in the lower part of Figure 15.4. Figure 15.21 shows a display created in GeoGebra (see Flashman [6] to explore the dynamic software on the internet). It is in three framed areas. The central area shows the mapping diagram for the function f (z) = z3 + pz + q from the z-plane to the w-plane for p = −1.9, q = 3.3. On the left is an area showing the z-plane and a circle radius δ = 1.0 with m = 24 points equally spaced around it. The area on the right shows the graph of the real function y = x3 + px + q. On the screen are boxes to input the formula for the function f and the variables p and q, as well as sliders with movable points to vary p, q, δ , and m. The central area can also be selected to change the viewpoint or the scale.

Figure 15.21 Using Geogebra to explore the cubic function f (z)

with this worksheet at www.cambridge.org/gb/S&T-geogebra

= z3 + pz + q. You can interact

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Inﬁnitesimals in Real and Complex Analysis

New possibilities arise that cannot be adequately described using a static picture in a textbook. The design provides two boxes to tick: one (MD) hides or shows the Mapping Diagram, which consists of the arrows from the points around the circle radius δ in the z-plane to the w-plane; the other shows or hides the roots of the equation f (z) = 0. The three roots marked r1 , r2, r3 are shown in the panel on the left, where r3 is on the real axis, and r1 , r2 are complex conjugates. Moving q up or down moves the real graph in the right frame dynamically up or down by the same amount. The graph as pictured has one real root near −2, and two complex conjugates near 1 ³ i. As q decreases, the real cubic curve moves down by the same amount until the function has three real roots. As this happens, the roots r1, r2 in the left frame move towards each other until they coalesce as a repeated real root, then separate into two distinct real roots, so that the cubic has three real roots. The user can manipulate the picture dynamically to focus on aspects of interest. For example, the left panel has an icon ´ ± which, when clicked, causes the chosen parameter (here p) to vary dynamically over its given range (here −5 to 5) and, as it does so, it displays the changing situation of the roots of the real cubic as the equation shifts between having one real and two complex conjugate roots to having three real roots. The GeoGebra environment allows the user to create and explore dynamic ﬁgures in complex analysis, Flashman [5].

15.12.5

Summary Interpretations of complex analysis continue to evolve into the future. New approaches in modern times are part of a more comprehensive overall picture in which intuition may be translated into rigour and conversely, proving structure theorems returns formal rigour to more sophisticated forms of symbolic manipulation and visual intuition. This broader vision reveals the fundamental distinction between real and complex analysis. Real analysis considers functions that exhibit many different properties and require highly sophisticated construction of the hyperreal numbers to cope with the logical introduction of inﬁnitesimals. Meanwhile, complex differentiable functions are given locally by power series around any point in a connected open domain and these can be manipulated in the much simpler extension ﬁeld Cε generated by a single inﬁnitesimal ε . This simpler extension encompasses the more powerful theory of analytic continuation and Riemann surfaces. From this higher vantage point we can see, at a single glance, the distinction between the technically complicated theory of real analysis and the subtle sophistication of complex analysis.

15.13

Exercises 1. Figure 15.22 shows the path γ : [ −π , π ] → C given by γ (t) = t + it and the analytic function w = f (z) = cos z. Calculate f γ , γ ± (t), and ( f γ )± (t) as functions of t. Interpreting t as time, consider the points γ (t) and ( f γ )(t) moving along the paths im γ , im f γ . By comparing the motion in the w-plane in the intervals −π ≤ t ≤ 0

15.13 Exercises

347

Figure 15.22 Geometry for Exercise 4.

and 0 ≤ t through 0.

≤ π , or otherwise, explain what happens in the w-plane as t passes

2. Let γ : [−1, 1] → C be given by γ (t) = t3 + it2 . Use a graph plotter to draw the parametric graph of (t 3, t2 ) and magnify the graph at various points. Conjecture what the highly magniﬁed portions of the graph will look like for t = 0 and t ² = 0. Calculate γ (t) numerically for t = −0.001, t = 0, t = 0.001 and draw the graph for −0.001 ≤ t ≤ 0.001 on a standard piece of paper. What happens if the graph is scaled from t = −10−6 to +10−6 on a standard piece of paper? 3. For f (z) = z2, plot the paths γ1(t) = 1 + it (−1 ≤ t ≤ 1), γ2(t) = −1 + it (−1 ≤ t ≤ 1), γ3(t) = it (−1 ≤ t ≤ 1) in the z-plane and their transformed paths f γ1 , f γ2, f γ3 in the w-plane where w = f (z). Imagine t increasing smoothly in time from t = −1 to t = 1 and trace the movement of the corresponding point along each graph with your ﬁnger. One has a different behaviour from the other two. Which one? Explain why, by considering the critical point of f . 4. Determine the Taylor expansion f (z0 + h) of the following functions about a point z0 and deduce the order of the variable part f (z0 + h) − f (z0 ) at that point: (i) sin z − z at z0 = 0 (ii) cos z at z0 = 0 and at z0 = i (iii) z5 at z0 = 0 and at z0 ² = 0 (iv) The principal value Logz at z = 1

5. Let K be a proper ordered ﬁeld extension of the real numbers R. For x ∈ K, write down the deﬁnition for x to be ﬁnite, positive inﬁnite, negative inﬁnite, inﬁnite, positive inﬁnitesimal, negative inﬁnitesimal, inﬁnitesimal. In each of the following cases, decide whether the given statement is true or false for all x, y ∈ K. If it is true, give a proof; if it is false, provide a counterexample (for instance, using the ﬁeld K = R(v) of rational functions in a variable v). (i) x negative inﬁnite implies 1/x is negative inﬁnitesimal (ii) x inﬁnite implies 1/ x is inﬁnitesimal (iii) x inﬁnitesimal implies 1/ x is inﬁnite (iv) x ﬁnite, y inﬁnitesimal implies xy is ﬁnite (v) x ﬁnite and non-zero, y inﬁnitesimal implies xy is inﬁnitesimal

348

Inﬁnitesimals in Real and Complex Analysis

(vi) x inﬁnitesimal, y inﬁnitesimal implies x/y is inﬁnitesimal (vii) x ﬁnite, y inﬁnite implies xy is inﬁnite

6. For x ∈ K[i] where i 2 = −1, write down the deﬁnition for x to be ﬁnite, inﬁnite, inﬁnitesimal. In each of the following cases, decide whether the given statement is true or false for all x, y ∈ K[i]. If it is true, give a proof; if it is false, provide a counterexample. (i) x inﬁnite implies 1/x is inﬁnitesimal (ii) ) x inﬁnitesimal implies 1/x is inﬁnite (iii) x ﬁnite, y inﬁnitesimal implies xy is ﬁnite (iv) x ﬁnite, y inﬁnitesimal implies xy is inﬁnitesimal (v) x inﬁnitesimal, y inﬁnitesimal implies x/y is inﬁnitesimal (vi) x ﬁnite, y inﬁnite implies xy is inﬁnite.

7. In the ﬁeld K = R(v) of rational functions in a variable v, place the following elements in linear order and in order of inﬁnitesimality: v, 1 − v, 1/ (1 − v), −1/v, v/(v − 1) 8. Prove Proposition 15.11 that if F is the subset of ﬁnite elements in a super ordered ﬁeld K and I is the subset of elements that are zero or inﬁnitesimal, then a, b ∈ F and c, d ∈ I implies a + b, a − b, ab ∈ F, c + d, c − d, cd

∈I

9. Prove Proposition 15.12 that if x, y ∈ F, where F is the subset of ﬁnite elements in a super ordered ﬁeld, then the standard part map st satisﬁes st (x + y) = st (x) + st (y),

st (x − y) = st (x) − st (y), st (xy) = st (x) st (y)

st (x/y) = st (x)/st (y) (for st (y)

²= 0).

10. Calculate the coefﬁcients cm +n, cm+n+1 , cm+n+2, cm+n+3 which arise when division

²∞ ±

r =m

in the superreal ﬁeld the equation

ar ε

r

³ ²∞ ±

/

r =n

brε

³

r

=

² ∞ ±

r=m +n

cr ε

³

r

Rε and the supercomplex ﬁeld Cε , is determined by solving

(cm+nε m+n +· · ·+ (cr ε r +· · · )(bn εn +· · ·+ (br ε r +· · · ) for successive coefﬁcients cm+n , cm+n+1 , . . ..

= (amε m +· · ·+(ar εr +· · · )

11. Show that there are some positive inﬁnitesimals δ ∈ Rε that fail to have a square root in Rε . Nevertheless, show that, for any z = x + iy ∈ Cε where x, y ∈ Rε , the º 2 2 modulus | z| = x + y can be calculated as an element in Rε . (Hint: each of x, y is either zero, or of the form aε n(1 + δ ), where δ is inﬁnitesimal.)

15.13 Exercises

349

12. For an inﬁnitesimal δ , calculate which of the following are ﬁnite and which are inﬁnite (using power series for trigonometric or exponential functions where necessary): (i) 3δ + 1/δ (ii) sin(iπ + δ ) (iii) δ 2 sin(1/δ) (iv) sin(iπ + δ ) (v) eiπ+δ (vi) eiπ+k δ for k ∈ C In those cases where the element is ﬁnite, calculate its standard part. 13. For f : D → C (where D is a domain in C) given by the following formulas, calculate the standard part of ( f (z + h) − f (z))/h in a super complex ﬁeld, where h is an inﬁnitesimal, and hence ﬁnd the derivative f ± (z) for z ∈ D. (i) z3 − 2z2 (ii) 1/(z − 2) for z ² = 2 (iii) (1 + z3 )n for n ∈ N (iv) sin z (in a super complex ﬁeld including power series expansions) (v) z2 sin z

14. Write down the deﬁnition for hyperreal numbers ∗ R, hypercomplex numbers ∗ C and the transfer principle for hyperreal and hypercomplex numbers. (i) Use the ﬁrst order statement ∀ x ∈ R ∃n ∈ N : n > x to deduce that there exist inﬁnite hypernatural numbers N ∈ ∗ N, where N ² ∈ N. (ii) By considering the property

∀m, n ∈ N : m > n ⇒ m − n ∈ N use the transfer principle to show that if m is an inﬁnite hypernatural number and n is a natural number, then m − n is an inﬁnite hypernatural number. (iii) Let E = {2n ∈ N : n ∈ N}, O = {2n − 1 ∈ N : n ∈ N} be the set of even and odd natural numbers. Use the transfer principle to show that every inﬁnite hypernatural number N is either even (in ∗ E) or odd (in ∗ O) and that N is even if and only if N + 1 is odd. 15. Using the hypercomplex numbers, by considering the standard part of zN for inﬁnite N, determine whether the sequence (zn) converges. If it does, calculate the limit, if not, explain why. (i) (n3 + sin n)/(n3 − 2n2 ) (ii) 1 + k + k2 + · · · + kn−1 for k ∈ C, k < 1 2n3 + 3n − 17 for n even (iii) 1 + 21 + ( 21 )2 + · · · + ( 21 )n−1 for n odd, 2 (n + 3n)(n + 1) n3 + sin n 1 (iv) 3 for n odd, 2 + for n even 2 n − 2n n 16. Visit the Geogebra website www.geogebra.org/ and use it to investigate some of the curves in C and R3 discussed in this book.

16

Homology Version of Cauchy’s Theorem

In Chapter 8 we proved the classical version of Cauchy’s Theorem, using step paths to simplify the geometry and topology involved. In Chapter 9 we developed the concept of homotopy, and reformulated Cauchy’s Theorem in that framework. Homotopy is one of the basic ideas in algebraic topology. A related topological concept, homology, emerged a little earlier from work on the classiﬁcation of surfaces. This notion and its extensions are of great importance in algebraic topology. When applied to domains in C, it provides fresh insight into Cauchy’s Theorem and its consequences. In complex analysis we view homology as a topological property of a domain D ⊆ C, but we deﬁne it in terms of properties of paths in D. There are then deep connections between the homology of D and the behaviour of complex integration along paths in D. In this respect homology is like homotopy, which is also a topological property deﬁned in terms of paths. The two concepts are related, but different. For homotopy, two paths in a domain are considered to be equivalent if one can be continuously deformed into the other. In the context of homology, two paths in a domain are considered to be equivalent if they differ by a path that forms the boundary ∂ R of a rectangle R whose image lies in D, as described in Section 9.3 – and, by extension, if they differ by a ﬁnite set of such boundary paths. In this chapter, we develop a homology version of Cauchy’s Theorem, and examine a few of its consequences, including a homology version of Cauchy’s Residue Theorem. The material here is a natural continuation from Chapter 9. We have postponed it until now to avoid delaying the more standard material in Chapters 10–14. The main ideas are simple, elegant, and geometrically intuitive, although the formal setting takes a little getting used to. Also, proofs have a habit of getting tangled up in combinatorial arguments about step paths. Here, algebra often comes to our rescue. The topology is fairly subtle, and results that seem obvious sometimes require a little care if we want a rigorous proof. We work throughout with step paths that lie in a domain D ⊆ C. Recall that a domain, in this context, is open and connected; moreover, being connected is equivalent to being path-connected. Cauchy’s Theorem concerns the integral of a complex function on D over a closed step path in D. 16.1. For the ﬁgures in the chapter, we often do not draw paths explicitly as step paths, because lots of little steps everywhere just make the ﬁgures more complicated without adding any understanding. We draw the steps only when they matter.

REMARK

Homology Version of Cauchy’s Theorem

16.0.1

351

Outline of Chapter We sketch the main ideas of this chapter now. Formal deﬁnitions and proofs come later. Throughout this chapter, we abbreviate ‘step path’ to ‘path’, except when we want to emphasise that we are working with step paths. Our terminology and approach to homology are tailored to the material in this text, and adapted to the simpler geometry of step paths. Topologists generally set up homology in a more general, more abstract, and more elegant manner, see for example Hatcher [8] or Hocking and Young [9]. Homology provides a natural context for several features of complex integrals that we have previously noticed:

• • • •

Integrals are additive: the integral of a given function along a sum of paths is the sum of its integrals along the separate paths. Moreover, the integral of the function along the reverse of a path is minus the integral along the path, allowing subtraction as well as addition. Cauchy’s theorem for a boundary, Theorem 9.6, states that if R is a rectangle and φ : R → D is continuous, and f : D → C is differentiable, then the integral of f along the boundary of φ vanishes. On the other hand, the integral of f along an arbitrary closed path need not vanish. Its value depends on how the path winds around points not in D.

All of these facts are closely related, suggesting that they can all be put together in a natural context. The question is: what? Our deﬁnition of a sum of paths in Section 2.8 involves joining them end to end, and we have to impose special conditions for that to be possible. This corresponds to the classical view that a function should be integrated ‘along’ a path. However, we can extend integration to any ﬁnite set of paths, even if they do not join up to create a single path, just by adding the separate integrals together. A path γ can be repeated any number of times to give a path nγ for n ∈ N \ {0}. For negative n we can deﬁne nγ to be −n times the reverse path −γ . Finally, the empty set of paths plays the role of 0. So we can deﬁne an arbitrary integer combination m1 γ 1 + · · · + mn γ n of paths γj , for mj ∈ Z . At ﬁrst glance, these combinations seem to form a group (indeed, an abelian group) under addition, but there is a technical problem: if γ is a path, γ + (−γ ) ought to be the zero element of this group, which is empty. Actually, as described above, it is the set {γ , −γ}, which is not empty. On the other hand, the integral of f along γ + (−γ ) is always zero, so this discrepancy has no effect on integrals. All of which suggests that we must somehow redeﬁne γ + (−γ ) to be zero. Mathematicians have a standard trick to resolve this kind of difﬁculty: introduce an equivalence relation in which γ + (−γ ) is equivalent to 0, and work with the equivalence classes. The equivalence relation we need is homology, and the clue for how to deﬁne it is Cauchy’s Theorem for a boundary. Essentially, we want any boundary path to be equivalent to zero. Everything else follows from that.

352

Homology Version of Cauchy’s Theorem

Since we are trying to construct a group, a natural way to achieve all this is to build in the group property right at the start. Then Cauchy’s Theorem for a boundary tells us how to deﬁne the equivalence relation of homology using group theory, where it takes a very simple, though more abstract, form. To accomplish this, we deﬁne integration over ‘chains’. A chain is a formal integer combination

± = m1 γ 1 + · · · + mn γ n where the γj are step paths (not necessarily closed) and the mj ∈ Z . We explain what this means, how to formalise it, and how to visualise it, in Section 16.1. When all mj = 0 the chain is empty, and this possibility is allowed in the deﬁnition. Be warned: addition here is not quite the same as addition of paths as deﬁned in Section 2.4, and −γ is not quite the same as the reverse path to γ . As we have just seen, the classical operations + and − run into technical problems, and we have to get round those. However, the classical notions and the abstract group operations are closely related. The precise relationship is given in Lemma 16.15: we can identify them ‘modulo homology’. The group of chains has two important subgroups. The ﬁrst is the subgroup of ‘cycles’, which correspond to formal sums of closed step paths, which we call ‘loops’. Inside the subgroup of cycles we deﬁne a further subgroup of ‘boundaries’. A boundary is now a formal integer combination

β = m1β1 + · · · + mr βr where each βj = ∂ R j is the boundary of a rectangle in D, and the mj ∈ Z. Every boundary of a rectangle is a loop, so in particular boundaries are cycles. By ‘rectangle in D’ we mean a continuous map φ : R → D such that the image φ (R) ⊆ D, just as in Cauchy’s Theorem for a boundary. Chains, in contrast, are deﬁned by the image φ (∂ R), and in that case the map is not required to be deﬁned on the interior of R, let alone to have its image a subset of D. Indeed, the distinction between these two cases is central to homology. To complete the set-up, two cycles ²1, ²2 are said to be ‘homologous’ if their difference ²1 − ²2 is a boundary. Another term for being a boundary is ‘homologous to zero’. The machinery of chains, cycles, boundaries, and homology is beautifully adapted to natural properties of complex integrals along paths. Let f : D → C be a continuous function. Integration over paths can be generalised to integration over chains, hence also over cycles or boundaries, by deﬁning

±

±

f

= m1

±

γ1

f

+ · · · + mn

±

γn

f

Boundaries come into play because we proved in Theorem 9.6 that the integral round the boundary of a rectangle whose image lies inside a domain is zero:

±

∂R

f

=0

Homology Version of Cauchy’s Theorem

353

So the integral of f over a cycle ² is unchanged if we add or subtract a series of boundaries of rectangles – that is, if we add a boundary to ². In other words, for any ﬁxed f , the integral depends only on the homology class of the cycle. We also need to deﬁne winding numbers for chains, which we do by deﬁning w(±, z0) = m1 w(γ1, z0) + · · · + mn w(γn, z0) We already know that the winding number for a path can be speciﬁed by an integral as in (7.6). This relationship also holds for chains. Once these concepts have been set up, we come to the main aim of the chapter, which is to prove: 16.2 (Homology Version of Cauchy’s Theorem). Let f : D → C be a continuous function, and let ² be a cycle in D. Then the following are equivalent: THEOREM

(i) ² is homologous to zero (that is, it is a boundary) in D. (ii) All integrals over ² vanish:

±

²

f

=0

for all differentiable f .

(iii) All winding numbers of ² around points outside D vanish: w(², z0 ) = 0

for all z0 ± ∈ D.

The proof of this theorem occupies most of this chapter. When the proof is complete, we derive some useful consequences.

16.0.2

Group-theoretic Interpretation Homology can be interpreted group-theoretically, as Emmy Noether noticed when attending lectures on topology by Heinz Hopf in 1926–7. Leopold Vietoris and Walther Mayer independently discovered the same interpretation at much the same time. Chains C , being formal integer combinations of speciﬁc objects (here paths), form an abelian group under formal addition. Cycles Z form a subgroup of C , and boundaries form a subgroup B of Z . Because the group of chains is abelian, these are normal subgroups. The equivalence relation of homology on Z determines a quotient group

H = Z /B = cycles/boundaries Speciﬁcally, the equivalence classes for homology are precisely the cosets of B in Z . All four groups C, Z , B , and H depend on D. The group H is the ﬁrst homology group of D, usually written H1(D). (In the topology of higher-dimensional spaces in Rn and Cn there are also ‘higher’ homology groups Hk (D) for k = 0, 1, 2, 3, . . ., but only H1 (D) concerns us here.) The elements of H can be thought of as the ‘essentially different’ topological types of loops in D, up to homology. With hindsight, our previous results related to Cauchy’s Theorem offer strong hints at the existence of this algebraic structure. As already noted, integrals vanish on boundaries. Paths can be added to give new paths, and integrals are additive. So are winding

354

Homology Version of Cauchy’s Theorem

numbers. Our aim is to develop a formal algebraic context that uniﬁes these hints. Algebraic topology originated from these and related considerations. To make the discussion as self-contained as possible, our approach will be classical (a polite way to say ‘old-fashioned’). The modern approach to homology is more general and more abstract.

16.1

Chains Let D ⊆ C be a domain. Recall that a (step) path in D is a map

γ : [a, b] → D

with the step property (sum of ﬁnitely many line segments parallel to the real or imaginary axis). To keep the terminology simple, we deﬁne: D E FI N I T I O N

γ (b).

16.3. A loop in D is a closed step path

Let

P

γ

: [a, b]

D, so

γ (a) =

= {all paths in D}

We turn P into an abelian group using an algebraic trick (which can turn any set into a subset of an abelian group): D E FI N I T I O N

16.4. A chain ± in D is a formal integer combination

± = m1 γ 1 + · · · + mn γ n where the γj ∈ P are distinct paths in D and the mj ∈ Z.

(16.1)

The set of all chains is denoted by C .

A more respectable, but less convenient, deﬁnition views the mj as the values of a function m : P → Z, which maps γj to mj . Thus we deﬁne

Z (D)

= {m : P → Z : m(γ ) = 0 for all except a ﬁnite number of γ ∈ P }

We can pass between functions m and formal sums (16.1) as follows: (i) Given (16.1), deﬁne m by: m(γ ) =

²

mj 0

if if

γ = γj γ ±= γj for any j

(ii) Given m, deﬁne

± = {γ ∈ L : m(γ ) ±= 0} mj = m( γj ) for γj ∈ ± We can add two such functions: (m1 + m2)(γ ) = m1 (γ ) + m2 (γ ) and we then have:

355

16.1 Chains

Figure 16.1 Visualising a chain. This is

P RO P O S I T I O N 16.5.

± = 3γ1 − 4γ2 + 2γ3 . It so happens that γ1 is a loop.

The set of chains C (D) is an abelian group under the operation +.

Proof. The zero element of C(D) is the function m for which m(γ ) = 0 for all γ The additive inverse of a function m is the function −m deﬁned by

∈ L.

(−m)( γ ) = −(m(γ ))

It is easy to see that + is associative and commutative. Technically, C is the ‘free abelian group’ generated by the set of paths P . At ﬁrst sight, it is not obvious how to visualise a formal sum of paths, especially one with integer coefﬁcients. One way is to draw the images of the different paths γj in the sum, and label each by the corresponding ‘multiplicity’ mj , as in Figure 16.1. ∑ The image of a cycle ± = mj γj is deﬁned to be the union of the images of the γj for those j with mj ± = 0. We extend the deﬁnition of the integral and the winding number from loops to chains: D E FI N I T I O N

16.6. Let

Then deﬁne

±=

mjγj be a chain, where the

(i)

±

=

³

w(± , z0 ) =

³

± (ii)

f

mj

j

j

±

γj

f

mj w(γj , z0 )

The deﬁnition trivially implies: P RO P O S I T I O N 16.7.

±

±1

f

+

±

± ±2

−±

±

f

=

f

=−

±±1 +±2 ±

f

f

γj are paths and z0 ∈ C.

356

Homology Version of Cauchy’s Theorem

Using Theorem 7.8 is it easy to prove: P RO P O S I T I O N

16.8. The winding number is additive: w(0, z0) = 0

w(±1 + ±2, z0) = w(±1 , z0 ) + w(±2, z0 ) w(−±, z0) = −w( ±, z0 )

We can now generalise Section 7.5:

16.9. Let ± = lie on the image of ±. Then

THEOREM

mj γj be a chain, and let z0 be any point in C that does not w(±, z0) =

Proof. w(± , z0 ) =

=

³ j

mj w(γj , z0 )

³ 1 j

= 21π i =

±

dz 2 π i ± z − z0 1

2π i

mj

³ ±

j

±

dz

γ j z − z0

± mj

dz γ j z − z0

dz 1 2π i ± z − z0

We sometimes need to make an explicit distinction between a path γ : [a, b] → D C and its image. To do so we write the image as

γˆ = {γ (t) : t ∈ [a, b]} We include the hat only when it might be confusing to omit it.

16.2

Cycles Next we need a special type of chain, which we call a cycle. A cycle is a ‘closed chain’. That is, the group of cycles is generated by chains whose images decompose into a series of closed loops. That is, if the usual sum γ1 + · · · + γn is a loop, then the corresponding formal sum in C is a cycle. Then all formal sums of cycles are again cycles. By deﬁnition the cycles form a subgroup of C , which we denote by Z . Next we deﬁne a subgroup B of Z , whose elements we call ‘boundaries’, which consists of cycles that can be ignored when calculating integrals (and therefore winding numbers). That is, cycles ² such that

±

²

f

=0

for all differentiable f : D → C

16.2 Cycles

357

(As a convention, we use ± , γ when discussing chains and ², ω when discussing cycles.) If we let 1 1 f (z) = (z0 ± ∈ D) 2π i z − z0 then Theorem 16.9 implies that all boundary cycles automatically also satisfy w(², z0 ) = 0

(z0

±∈ D)

Which cycles should we use? By Theorem 9.4 the group B should include all boundaries ∂ R of rectangles R, as in Section 9.3. Several special loops of this type are particularly useful.

16.2.1

Sums and Formal Sums of Paths We have already deﬁned γ +δ and −γ for paths: respectively, adjoin the two paths and combine them into a single one, and reverse the path. These operations are not those of C . However, we want them to be the same ‘up to homology’, see Section 16.4. That is, they should be equal modulo B. We can then think of these geometric operations as being the corresponding operations in C (or the subgroup Z , which is more important here) whenever homologous chains can be considered equivalent. Recall from Section 6.8 that if γ : [a, b] → D and δ : [b, c] → D (not necessarily closed paths) we deﬁned −γ : [a, b] → D and γ + δ : [a, c] → D by

−γ (t) = γ (b + a − t)

and (γ

² (t) + δ)(t) = γδ(t)

if t if t

∈ [a, b] ∈ [b, c]

(The original deﬁnition was slightly more general, allowing the interval for δ to be [c, d] and then shifting it to make c = b. That option is not needed here.) See Figure 16.2. Although we are using the same symbols + and −, these operations are not (quite!) those of the group C , and we need to sort out the precise relationship. To avoid confusion, we temporarily write the operations in C as ⊕ and ², and use + and − for the operations on paths that we deﬁned in Chapter 6. First: they really are different. The path-sum γ + δ , considered as an element of C , corresponds to the function m : P → Z such that m(γ

Figure 16.2 Previous deﬁnitions of

thought of as cycles.

+ δ) = 1

−γ and γ + δ. Here both paths are loops, so they can also be

358

Homology Version of Cauchy’s Theorem

In contrast, γ

m(σ ) = 0

±= γ + δ)

⊕ δ ∈ C corresponds to the function m³ : P → Z such that m³ (γ ) = 1 m³ (δ ) = 1 m³ (σ ) = 0 (σ ± = γ , δ ) These are different functions because, as elements of P , we have γ ⊕δ ± = γ , γ ⊕ δ ± = δ ,

so

m³( γ

⊕ δ) = 0 m(γ + δ) = 1 Similarly, ²γ (cycle) is not the same as −γ (path). 16.3

Boundaries The role of homology is, in effect, to force these technically different forms of addition and reversal to be the same. It exploits a standard trick encountered throughout mathematics: deﬁne an equivalence relation for which objects that differ in inessential ways are equivalent. Then their equivalence classes are the same. For instance, in arithmetic modulo n, a difference between integers that is divisible by n is inessential. We would like to set n = 0, but that is technically false. So we deﬁne r, s to be congruent modulo n if r − s is divisible by n. Congruence is an equivalence relation, and the set of equivalence classes is denoted Zn . Now, although r and s are different in Z, they represent the same congruence class. In this case Zn is the quotient group of the additive group of integers Z by the subgroup of nZ of multiples of n; that is, Zn = Z/ (nZ). (The multiplicative structure can also be included, because nZ is an ideal of the ring Z .) We can play the same game with cycles in place of Z , and whatever differences we wish to ignore playing the role of nZ. In this case, the differences that we want to forget about are boundaries, in the following sense: 16.10. The group B of boundaries in D is the subgroup of by all boundaries ∂ R of rectangles in D.

D E FI N I T I O N

Z generated

To make the argument clear, we continue to use ⊕, ² in place of +, − in C to distinguish the group operations from the standard operations on paths. To get round one further technicality we go the whole hog and throw in all reparametrisations of γ that preserve orientation. Consider an orientation-preserving homeomorphism ρ : [a, b] → [a, b]. That is, ρ is continuous and has a continuous inverse ρ −1, and ρ(a) = a, ρ(b) = b. Equivalently, ρ is monotonic strictly increasing on [a, b] and satisﬁes ρ (a) = a, ρ (b) = b. Now γρ (t) = γ (ρ (t)) deﬁnes a loop γρ . By Proposition 9.15, this change has no effect on integrals, hence on winding numbers. If we allow ρ to reverse orientation, the integral changes sign, which is why we require orientation to be preserved. An alternative is to allow orientation to be reversed as well. Then −γ is just a special reparametrisation, and any orientation-reversing

16.3 Boundaries

359

Figure 16.3 Maps of rectangles used in the proof of Proposition 16.11. Top: Reversal. Middle:

Sum. Bottom: Reparametrisation.

reparametrisation is the map reparametrisation.

γ ´→ −γ

composed with an orientation-preserving

P RO P O S I T I O N 16.11. Let γ , δ be paths in D and let reparametrisation of γ . Then

γρ be an orientation-preserving

γ ⊕ (−γ ) ∈ B (16.2) (γ ⊕ δ ) − (γ + δ ) ∈ B (16.3) γ ² γρ ∈ B (16.4) Proof. In each case we specify a map φ : R → D with a suitable boundary. Figure 16.3 illustrates the underlying geometry. Assume that γ : [a, b] → D and δ : [b, c] → D; also that ρ : [a, b] → [a, b] is the reparametrisation. For (16.2), let (t, s) ∈ [a, b] × [0, 1] = R and set φ (t, s) = γ (t) (so φ is independent of s). Then φ (R) = γ ([a, b]) ⊆ D, and ∂φ = γ + (−γ ). For (16.3), let (t, s) ∈ [a, c] × [0, 1] = R and set ² if t ∈ [a, b] φ (t, s) = γδ(t) (t) if t ∈ [b, c] Now φ(R) = γ ([a, b]) ∪δ ([c, d]) ⊆ D, and ∂φ can be decomposed into ( γ ⊕δ ) ² (γ +δ ). For (16.4), let (t, s) ∈ [a, b] × [0, 1] = R and set φ(t, s) = γ ((1 − s)t + sρ(t))

360

Homology Version of Cauchy’s Theorem

Then φ (R) = γ ([a, b])

⊆ D, and ∂φ = γ − γρ . P RO P O S I T I O N 16.12. If ² ∈ B then ±

² and

f

=0

for all differentiable f : D → C w(², z0 ) = 0

(z0

±∈ D)

Proof. It is enough to check that these conditions hold for boundaries of rectangles in D, because both the integral and winding number are additive by Propositions 16.7 and 16.8.

16.4

Homology The rather pedantic distinction between ⊕ and +, and the other items discussed above, does not cause any great problems, because – as we now prove – it disappears when we pass to homology classes. But ﬁrst, we must deﬁne homology: D E FI N I T I O N

16.13. Two cycles ²1 , ²2 in D are homologous, written ²1 ∼ ²2 , if

²1 ² ²2 ∈ B

The homology class of ² is its ∼-equivalence class: [²] = {²³ : ²³

∼ ²}

It is easy to see that [ ²] is the coset of B in Z that contains D E FI N I T I O N

². We therefore deﬁne:

16.14. The ﬁrst homology group of D is the quotient group H1(D)

= H = Z (D)/B (D)

It is abelian since Z (D) is abelian. We denote the identity element of LEMMA

H by 0.

16.15. For all loops γ , δ and orientation-preserving reparametrisations ρ ,

γ ⊕δ ∼ γ +δ ²γ ∼ −γ γρ ∼ γ

Proof. In each case the left- and right-hand sides differ by an element of Proposition 16.11. Therefore they deﬁne the same element of Z /B = H.

B,

by

This lemma shows that the distinction between ⊕ and +, and the other similar distinctions, disappear when we work modulo B. We therefore use the ordinary symbols +, − for cycles as well as loops, and stop using ⊕, ². They have served their purpose.

16.4 Homology

∼ γ ³ and δ ∼ δ³ then γ + δ ∼ γ ³ + δ³ . Both γ − γ ³ and δ − δ ³ lie in B. Now add.

LEMMA

Proof.

361

16.16. If γ

R E M A R K 16.17. Homology ‘really’ works with the image γˆ of a path rather than the path γ : [a, b] → D. More precisely, what matters is the image and its orientation. Orientation is relevant because γ and −γ have the same image, but the integral along one is minus that along the other. We want γ − γ ∈ B but not γ + γ ∈ B to make integrals vanish on cycles in B.

Example 16.18. Let D

and δ : [0, 1]

= C \ {0}. Deﬁne γ : [0, 1] → D by γ (t) = e2π it

→ D by

δ(t) = 2e2π it

as in Figure 16.4 (left). We claim that γ ∼ δ . To prove this, introduce the path σ = [1, 2] and its reverse −σ = [2, 1], each parametrised by t ∈ [0, 1]. Now γ − δ is homologous to γ + σ − δ − σ because modulo B we have

γ + σ − δ − σγ − δ + σ − σ = γ + δ + 0 = γ + δ Then we claim that γ + σ − δ −σ = 0 modulo B . This follows because γ +σ − δ − σ is the boundary of a rectangle. Speciﬁcally, let R = [0, 1] × [0, 1] and deﬁne a map φ : [0, 1] × [0, 1] → D by

φ(r, s) = (1 + s)e2π ir Then it is straightforward to check that ∂φ = γ + σ − δ − σ , see Figure 16.4 (right). But now, modulo B, we have γ − δ = 0, so γ ∼ δ .

16.18. It is possible to deﬁne the relation of homology using winding numbers. (Obtaining the homology group is less straightforward.) In this approach a boundary is deﬁned to be any cycle ² such that w(², z0) = 0 when z0 ± ∈ D. The proof of Cauchy’s Theorem still requires most of the same ideas, and we feel that this deﬁnition of homology loses contact with the original geometric ideas that inspired it. Eventually we will show that our deﬁnition is equivalent to the one based on winding numbers, but we have to prove Theorem 16.2 to do this. REMARK

The following result is crucial:

362

Homology Version of Cauchy’s Theorem

Figure 16.4 Left: The paths

γ + σ − δ − σ. THEOREM

in D. If ²1

γ , δ , σ , −σ . Right: Mapping a rectangle so that its boundary is

16.19. Let f : D → C be differentiable on a domain D. Let ²1 , ²2 be cycles

∼ ²2 then

±

²1

f

=

± f

²2

Proof. Since ²1 ∼ ²2 is equivalent to ²1 − ²2 statement is equivalent to If ² ∈ B then

∈ B, and the integral is additive, the

±

²

f

=0

Again using additivity, this follows if the integral over every generator of B is zero. But B is generated by boundaries of rectangles whose images lie in D, and the integral over the boundary of a rectangle is zero by Cauchy’s Theorem for a Boundary, Theorem 9.6. The winding number is a homology invariant: C O RO L L A RY

16.5

16.20. If ²1

∼ ²2 then w( ²1 , z0 ) = w(²2 , z0 )

for all z0

±∈ D

Proof of Cauchy’s Theorem, Homology Version We now give the proof of Theorem 16.2, the homology version of Cauchy’s Theorem. Cauchy’s Theorem for a Boundary, Theorem 9.6, gives the implication (i) ⇒ (ii). That (ii) ⇒ (iii) follows from Theorem 16.9. So it remains only to prove that (iii) ⇒ (i). Recall that condition (i) states that ² is homologous to zero in D, or equivalently, ² ∈ B . Condition (iii) states: w(², z0)

=0

for all z0

±∈ D

which we call the non-winding condition. Most of the proof that (iii) ⇒ (i) is an exercise in making careful distinctions between paths and their images, and using the relatively simple geometry of step paths to control

363

16.5 Proof of Cauchy’s Theorem, Homology Version

the topology. This does involve some rather pedantic distinctions, needed only for this proof. Let

²=

³

n j ωj

(16.5)

for loops ωj in D, and nj ∈ Z . We may assume that the ωj are distinct and that all nj ± = 0. (If not, ² = 0 ∈ B anyway.) We assume the non-winding condition and deduce that ² is homologous to zero. The idea of the proof is to simplify ² by repeatedly changing it to some cycle ²³ that is homologous to ², and to use induction on some suitable measure of the complexity of the cycle. The deﬁnition of ²³ is chosen so that the complexity decreases at each stage. A simple result along these lines is Corollary 16.20, which implies that if ² ∼ ²³ and z0 ± ∈ D, then w(², z0 ) = w(²³ , z0 )

Thus, if ² satisﬁes (iii) then so does ²³ . Trivially, if ²³ satisﬁes (i) then so does ². Therefore, if (iii) implies (i) for ²³ , the implication also holds for ². In other words: When proving (iii) implies (i) for ² we can replace it by any homologous ²³ .

16.5.1

Grid of Rectangles To construct suitable ²³ we proceed as follows. ˆ As in the proof of Lemma 8.6, we form a grid of rectangles. Extend each edge of ² (horizontal or vertical) to a line in C. These lines divide C into ﬁnitely many rectangles, and the lines cross at grid points, see Figure 16.5. Some of the ‘rectangles’ are inﬁnite; these play no role as we shortly see. ˆ , along with any corners or ‘ends’ where Let P be the set of all grid points that lie on ² ˆ terminates without continuing. (The corresponding path ‘doubles back’ at an edge of ² such a point. In the generality employed here, this can happen.)

Figure 16.5 Rectangles and grid points. Here

² is a sum of three loops.

364

Homology Version of Cauchy’s Theorem

We use P to split the ωj into sums of paths, such that the image of each path is an edge of some rectangle. Let the parameter values of ωj be t ∈ [a, b]. Since ωj is a loop we can change parameters if necessary so that ωj (a) ∈ P. (Start at a corner or some other grid point, and note that a change of parameter is a homology.) Then the set of t for which ωj (t) ∈ P (where the path goes through a point in P) is a ﬁnite set, and can be arranged in order as a = t0 < t1
0. Deﬁne A to be the top right point of P. That is, (i) im A ≥ im B for all B ∈ P (ii) re A ≥ re B for all B ∈ P such that im B = im A Since P is ﬁnite its top right point exists and is unique. We consider the geometry of ² ‘near A’. There are three possibilities:

ˆ but not on any horizontal edge. (i) A lies on a vertical edge of ² ˆ but not on any vertical edge. (ii) A lies on a horizontal edge of ² ˆ and on a vertical edge. (iii) A lies on a horizontal edge of ²

366

Homology Version of Cauchy’s Theorem

Figure 16.7 Three possible cases for the geometry near A. Dotted lines indicate possible

connections to the rest of the cycle.

Figure 16.8 Rerouting segments to get rid of R.

Figure 16.7 shows these three possibilities, where A, B, D ∈ P. In cases (i) and (ii), there must be a segment BA in that orientation, and another with the reverse orientation AB, because ² is a cycle. These two segments cancel, contrary to our previous assumption, so case (i) cannot occur. Similarly, case (ii) cannot occur. So only (iii) is possible. Orient AB from A to B and DA from D to A. Then all segments in ² with image AB occur m times, where 0 ± = m ∈ Z . If m > 0 they have orientation AB; if m < 0 they have orientation BA. Since ² is a cycle, the segments of ² with image DA must also occur m times, where 0 ± = m ∈ Z. If m > 0 they have orientation DA, of m < 0 they have orientation AD. (Any loop that arrives at A must leave A.) Let C be the fourth corner of the rectangle R = ABCD. Then A, B, C, D are grid points and A, B, D ∈ P. The geometry on R is shown in Figure 16.8 (left), with m indicating |m| copies of a segment in the direction of the arrow for m > 0, and in the opposite direction for m < 0.

16.5.3

Rerouting Segments We now claim that by using a homology we can re-route the segments of ² that lie in DAB to corresponding segments in DCB, as in Figure 16.8 (right). To set this up, we pause from the proof of Theorem 16.2 to prove: LEMMA

16.24. With the above notation, if z lies in the interior of R then w(², z)

=m

(16.6)

Proof. Draw a vertical ray λ from z upwards, to inﬁnity. This cuts AB at one (interior) point X, as in Figure 16.9 .

16.5 Proof of Cauchy’s Theorem, Homology Version

λ

Figure 16.9 Rectangle R and ray . Shaded area indicates half-plane containing

367

²2 .

Let ²1 be ² with all segments in DA and AB removed (this is not a cycle). Let ²2 be ²1 plus the rerouted segments in DC and CB (all of multiplicity m). Then ²2 is a cycle. We claim that

²2 ∼ ² Segments of ² lying in AB cut λ m times (with AB oriented to the left). The cycle ²2 does not intersect λ because A is the top right point of ², so ²2 lies in the half-plane below AB. By continuous choice of argument, as in Section 7.4, w(²2, z)

=0

(16.7)

Let

δ = m ∂R ∈ Z Clearly

² = ²2 + δ because segments of the right-hand side on DC or CB cancel, and the other segments of δ restore the segments on DA and AB. We now face a minor technical problem: we do not (yet) know that DC and CB lie in the domain D. We therefore need to prove this, as follows. If z is in the interior of R, then w(δ , z) = m by continuous choice of argument. AB crosses λ only at X, where it crosses m times in the anticlockwise direction. Therefore if z is in the interior of R, w(², z) = w(²2 + δ , z) = w(²2, z) + w(δ , z) = 0 + m using (16.7) and additivity of w.

= m ±= 0

368

Homology Version of Cauchy’s Theorem

16.5.4

Resumption of Proof of Theorem 16.2. By (iii), which we are assuming, we have z ∈ D. Therefore the interior of R lies in D. ˆ then BC ⊆ D. Otherwise, continuity of the winding number at points not If BC ∈ ² ˆ in ² implies that w(², z) = 0 if z lies on the interior of BC. The same goes for CD. ˆ then z ∈ D. If not, we again appeal to Finally, suppose that z = C. If C lies in ² continuity of w to deduce that w(², z) = m ± = 0. As above, z ∈ D. So the whole of R, interior plus boundary, lies in D. But now

δ = m ∂R ∼ 0

since ∂ R is a boundary, so lies in B. We ﬁnally conclude that

²2 ∼ ² − δ ∼ ²

The complexity c(²2 ) < c(²). The number of segments does not change, but we have removed the rectangle R. This is relevant since w(², zR ) = m ± = 0, where zR is the centre of R. On the other hand, it is easy to see that no other rectangle can be irrelevant for ²2 but relevant for ². The reason is that if zk is the centre of rectangle R k , the continuous argument deﬁnition of w implies that w(Rk , zl ) =

²

1 0

if if

k k

=l ±= l

Additivity then proves that for all rectangles Rk except R we have w(²2 , zk ) = (², zk ) Thus we have removed one relevant rectangle but not introduced any new ones, and c(²2) < c( ²) as claimed. Inductively, ²2 ∼ 0. But ² ∼ ²2 so ² = 0. This completes the induction step, proving Theorem 16.2.

16.6

Cauchy’s Residue Theorem, Homology Version We can now use Theorem 16.2 to prove a homology version of Cauchy’s Residue Theorem, Theorem 12.3. First: LEMMA

16.25. Let ²1 , ²2 be cycles in D such that w(²1 , z)

= w(²2, z) for all z ±∈ D

Then Proof. For all z ± ∈ D,

²1 ∼ ²2 w(²1 − ²2, z)

= w(²1, z) − w( ²2 , z) = 0 By Theorem 16.2, ²1 − ²2 ∼ 0. So ²1 ∼ ²2 .

369

16.6 Cauchy’s Residue Theorem, Homology Version

be

Suppose z1, . . . , zk are distinct points in C, and deﬁne the k-times punctured plane to D

= C \ {z1 , . . . , zk }

It is easy to see that D is a domain. (Recall: this means open and connected.) THEOREM

16.26. If D is the k-times punctured plane, then H1 (D) ∼ = Zk

= Z⊕ ··· ⊕Z

(k times)

Proof. We do more: we ﬁnd independent generators for H1 (D); that is, elements such that any element is a unique integer combination of them. ˆ. Let ² be a cycle in D. Then for j = 1, . . . , k, no zj lies on ² ˆ Since ² is a closed set, its complement is open, so there exists ε > 0 such that for all j = 1, . . . , k

ˆ =∅ Nε (zj ) ∩ ²

Let R j be a square (with horizontal and vertical sides) of side ε/2 centred at zj . Then ˆ = ∅. (We use a square to make ∂ Rj a step path.) Rj ⊆ Nε (zj ) so R j ∩ ² Deﬁne cycles

ρj = ∂ Rj

oriented anticlockwise. Then clearly

²

w(ρj , zl ) =

1 0

if if

j j

=l ±= l

(16.8)

using a suitable cut plane and a continuous choice of argument. Let mj We claim that This follows because

= w(², zj )

²∼ w(², zl ) = ml

³

mj ρ j

=w

´³

mj ρj , zl

µ

by additivity and (16.8). Now apply Lemma 16.25. We also claim that the mj are unique: no other Z-linear combination of the homologous to ². Equivalently, we must show that if

ρ=

³

mj ρj

∼0

ρj

is

(16.9)

then all mj = 0. Suppose (16.9) holds. Then w(ρ , z) = 0 for all z ± ∈ D by Theorem 16.2. That is, w(ρ , zl ) = 0 for all l = 1, . . . , k. But by Proposition 16.8, w(ρ , zl ) Thus ml

= 0 for all l = 1, . . . , k.

=w

´³

mj ρj , zl

µ

= ml

370

Homology Version of Cauchy’s Theorem

Clearly every ρ is a cycle, so the map

² = (m1 , . . . , mk ) induces an isomorphism H1(D)

→ Zk

We say that the homology class of ² is the corresponding (m1, . . . , mk ) ∈ Zk . 16.27 (Cauchy’s Residue Theorem, Homology Version). Let D be a k-times punctured plane and suppose that f : D → C is differentiable. Let ² by a cycle in D of homology class (m1 , . . . , mk ) ∈ Zk . Then THEOREM

±

² Proof.

²∼

∑k

ρ

j=1 mj j ,

= 2π i

f

k ³ j =1

mj res (f , zj )

so

±

²

f

=

k ± ³ j =1

ρj

f

Since f is differentiable, we can apply Theorem 12.3, Cauchy’s Residue Theorem for a loop, to obtain

±

ρj

f

= 2π i res (f , zj )

We could go on to state a more general version of Cauchy’s Residue Theorem for other domains, such as C with a discrete inﬁnite set of points removed, or a disc with ﬁnitely many points or discs removed. We could also consider analogues for Riemann surfaces. But it seems sensible to stop here, having opened up the possibilities of homology and used it to prove two powerful theorems in complex analysis.

16.7

Exercises 1. Let D = {z ∈ C : 1 < | z| < 2}. Prove that H1(D)

∼ = Z.

2. Let D be an open disc with k distinct points removed. What is H1 (D)? Prove your answer correct. 3. Let D be C with k disjoint closed discs removed. What is H1(D)? Prove your answer correct.

4. Suppose that K = {zk : k ∈ N} is an inﬁnite set of distinct points, and that K is discrete; that is, for each k there exists ε > 0 such that Nε (zk ) ∩ K = {zk }. Prove a version of Cauchy’s Residue Theorem for the domain D = C \ K. 5. Construct a closed step path in C \ {0} that is homologous to zero, whose complement has four distinct connected components.

16.7 Exercises

371

6. The 0th homology group H0 (D) can be deﬁned in a manner analogous to that for H1(D), as follows. A 0-cycle is a ﬁnite formal sum of pairs of (not necessarily distinct) points in D. A 0-boundary is a ﬁnite formal sum of pairs of points in D that can be joined by a step path in D. Show that 0-cycles form a group Z0 and 0-boundaries from a group B0 . Deﬁne H0(D) = Z0/ B0. Show that the pair (z1 , z2) is a 0-boundary if and only z1 and z2 lie in the same connected component of D. Deduce that H0(D) is generated by all cosets of B of the form (zK , zK ) + B , where one point zK ∈ C is chosen for each connected component K. (That is, H0(D) is isomorphic to the free abelian group on the connected components.) In particular, if D has a ﬁnite number k of connected components, prove that H0 (D) ∼ = Zk. 7. Prove the Dog Walking Theorem: If σ1, σ2 are loops, both parametrised by t ∈ [a, b], and there is a point z0 such that

|σ1 (t) − σ2(t)| < |σ1 (t) − z0 |

for all t ∈ [a, b]

then w( σ1, z0 ) = w(σ2, z0) and σ1 ∼ σ2 in C \ {z0 }. Explain the name of this theorem. (Hint: σ1 is the man, tree.) Use it to give another proof of Rouché’s Theorem 12.14.

σ2 the dog, and z0 is a

8. Jordan Contour Theorem for Loops. Recall that a loop is a closed step path. Say that a loop σ : [a, b] → C is simple if it does not cross itself; that is, σ (t1 ) = σ (t2 ) with t1 ± = t2 ∈ [a, b] only when the tj equal a or b. Use the trick in the proof of Theorem 16.2, of rerouting the segments of a cycle (here one loop) round the rectangle R adjacent to the top right point of σˆ to prove that: (i) C \ σˆ has exactly two connected components: one bounded, the other unbounded. (ii) Let I(σ ) be the bounded component in (i) and O(σ ) be the unbounded component. Prove that w(σ , z) = µ1 if z ∈ I(σ ) and w(σ , z) = 0 if z ∈ O( σ ). (iii) ∂ I(σ ) = ∂ O(σ ) = σˆ . 9. If σ1, σ2 are simple loops and O(σ2 ).

σˆ 1 ⊆ I(σ2), prove that I(σ1) ⊆ I(σ 2) and O(σ1 ) ⊇

10. Suppose that a simple loop σ lies in a domain D. Prove that there exists ε > 0 such that |z − σ (t)| < ε for z ∈ C and some t ∈ [a, b] implies that z ∈ D. (Hint: use the Paving Lemma.)

11. Using the result of Exercises 9 and 10, prove that if σ is a simple loop in D there exists ε > 0 such that there are simple loops σ I , σ O such that:

372

Homology Version of Cauchy’s Theorem

(i) Every point of σˆ O ∪ σˆ I lies within distance ε of some point of σˆ . (ii) σ O lies in O(σ ) and σ I lies in I(σ ). (iii) The regions I(σ O ) \ (σˆ ∪ I(σ )) and I(σ ) \ ( σˆ I ∪ I(σ I )) are connected. 12. If D is a domain (hence connected) and D ∩ O(σ ) and D ∩ I(σ ) are connected.

σ

is a simple loop in D, prove that both

13. Deﬁne the edges of a loop as follows. Deﬁne a corner to be a point at which the image of the loop switches between horizontal and vertical. Assume that σ (a) is a corner, and let Q be the set of t ∈ [a, b] such that σ (t) is a corner. Arrange the elements of Q in order: a = t0

< t1 < · · · < tn = b

Then an edge is the image of σ restricted to some interval [tj −1, tj ] for j = 1, . . . , n. Deﬁne the edges of a cycle to be those of its component loops. Say that a cycle σ is in general position if no two edges are collinear (that is, both lie in the same horizontal or vertical line in C). Prove that every cycle in a domain D is homologous to a cycle in general position. (Hint: inductively displace successive edges parallel to themselves through a small distance, so that the displaced edge differs from the original edge by the boundary of a rectangle lying inside D. Fix up the geometry at corners, where as stated there may be a gap or a small overlap between the displaced edges.)

14. Show that any non-zero cycle is homologous to a sum of disjoint simple loops σj , each occurring in the sum with multiplicity µ1. (Hint: use Exercise 13 to put it in

Figure 16.10 Left: Partial ordering of loops (here shown as ellipses, not step paths, for clarity) by containment of interiors. Right: Cancelling adjacent oppositely oriented loops by making a cut.

16.7 Exercises

373

general position, then reroute edges near crossing points to eliminate the crossings, using small rectangles inside D.) 15. Using induction on the number of simple loops in such a decomposition, obtain a different proof of Theorem 16.2, which is modelled along the classical lines of simplifying loops by ‘making cuts’ that cancel each other out. See for instance Figure 11.2. Warning: This exercise is hard, but the following sketch may help. Think of it as a mini-research project. First, show that disjoint simple loops are partially ordered by the relation σ1 < σ2 ⇐⇒ I( σ1) ± I(σ 2). This ordering can be complicated, see Figure 16.10. Show that if (iii) holds then there must be two simple loops σ1 < σ2 with opposite orientations, such that there is no simple loop σ 3 with σ1 < σ3 < σ2 . Using some of the exercises above, show that there is a step path τ joining a point on σ1 to a point on σ2 that lies in D and intersects no other σj appearing in the decomposition. Use τ to write

σ1 + σ2 = ρ + τ ³ where ρ is a simple loop in D and τ ³ is a simple loop in D that is homologous to

zero, see Figure 16.9. This reduces the number of simple loops by 1, and induction reduces to the case of one simple loop σ . Show that I(σ ) lies in D and complete the proof using property (iii).

17

...

The end of this book is fast upon us, but there is as yet no discernible end to complex analysis itself. It remains a vigorous and growing part of mainstream mathematics. We sample this rich area by giving brief descriptions of ﬁve areas of current research: the Riemann Hypothesis, modular functions, several complex variables, complex manifolds, and complex dynamics.

17.1

The Riemann Hypothesis One of the more unexpected applications of complex analysis occurred in the second half of the nineteenth century, leading to major advances in number theory. At ﬁrst sight these two areas have little connection: complex analysis is about continuous quantities and number theory is discrete. Even more surprisingly, the connection that emerged did so in the context of statistical properties of prime numbers. The starting point was a new kind of complex function, based not on power series, but on Dirichlet series of the form

φ(s) =

∞ a ± n n=1

ns

where the an and s are complex numbers and the domain is restricted to make the series converge. For example, if the an are bounded, we require re s > 1. More strongly, if the partial sums a1 + · · · + an are bounded, we require only re s > 0. Classical concepts such as the gamma function (and, later, more sophisticated new ideas) were then used to extend the deﬁnition of φ to the entire complex plane, except for isolated poles. Prime numbers are of vital importance in mathematics, yet they notoriously lack any clear pattern. Given a list of all primes up to some particular number, there seems to be no simple way to predict the next one. However, hints of regularity emerge when we think about entire ranges of primes. For instance: how many primes are there up to some speciﬁed limit x? It is difﬁcult to answer this question exactly, but good approximations are another matter. In 1797–8 Adrien-Marie Legendre counted how many primes occur up to various limits, using tables of primes provided by Jurij Vega and Anton Felkel. Denoting the number of primes less than x by π (x), as we now do, he observed empirically that π (x) seems to be close to x/(log x − 1.083 66). In a letter of 1849, Gauss wrote that when he was about 15 he noticed that π (x) is approximately x/ log x for large x. In 1838 Dirichlet wrote to Gauss to tell him he had found a similar but more accurate approximation to π (x), the logarithmic integral

17.1 The Riemann Hypothesis

Li(x) =

² 0

x

375

dt log t

For example, when x = 109 ,

π (x) = 50 847 534 Li(x) = 50 849 234.9 x/ log x = 48 254 942.4 The approximations here are asymptotic: that is, the ratio of the approximate formula to the true value π (x) was conjectured to tend to 1 as x → ∞. The error in an asymptotic formula can be quite big; it just has to be signiﬁcantly smaller than the exact quantity. All of these observations were purely empirical, and a rigorous proof that these formulas are asymptotic to π (x) became a major open problem in number theory, called the Prime Number Theorem (at a time when ‘theorem’ was also used to refer to a conjecture). The eventual proof of the Prime Number Theorem relied on deep applications of complex analysis. The relation between these apparently disconnected areas of mathematics goes back to Euler, who realised in 1737 that uniqueness of prime factorisation has an implication for real analysis, namely the formula

∞ 1 ± n=1

= ns

³ p

1 1 − p−s

where p runs through the primes. To prove this, observe that summing a geometric series gives 1 1 − p−s

= 11s + p1s + p12s + · · ·

Expand the product on the right-hand side, and observe that each term 1 /ns appears exactly once as 1 divided by the sth power of a product of prime powers. Euler considered only positive integer s, but in 1848 and 1850 the Russian mathematician Pafnuty Chebyshev attempted to prove the Prime Number Theorem by applying real analysis to Euler’s series, assuming s to be real and greater than 1, so that the series converges. He managed to prove that for large enough x the ratio of π (x) to x/ log x lies between two constants: one slightly bigger than 1, the other slightly smaller. Riemann realised that real analysis was too limited to complete the proof, but complex analysis was more powerful and might do the trick. He noticed that Euler’s series converges for any complex s with re s > 1. Its sum, called the zeta function ζ (s), is complex analytic. He wrote up his ideas in 1859. They included an explicit formula expressing π (x) exactly in terms of the zeta function. To state it, Riemann deﬁned a related function

±(x) = π (x) + 21 π (x1/2 ) + 31 π (x1/3 ) + 14 π (x1/4) + · · · which counts the prime powers up to x. From this it is easy to deduce proved that

π (x). Then he

376

The Road Goes Ever On . . .

±(x) = Li(x) −

±

ρ

Li(xρ ) +

² ∞ x

t(t2

dt − 1) log t

where the sum is over all zeros of the zeta function, excluding negative even integers. (We explain where those come from in a moment – their role is not apparent in the formula for the inﬁnite series.) The zeros and poles of a complex analytic function are important because they characterise the function completely if we also know their orders. It turns out that the zeros of the zeta function include all negative even integers, along with inﬁnitely many others. In 1896 Jacques Hadamard and Charles Jean de la Vallée-Poussin independently proved the Prime Number Theorem using Riemann’s zeta function, a triumph for what became known as analytic number theory – the application of complex analysis to properties of whole numbers. This approach has now blossomed into a huge area of number theory. Riemann took the crucial step of using analytic continuation to extend the deﬁnition of ζ (s) so that s can be any complex number except 1. This step depends on the gamma function, introduced in Section 6.11, which is a generalisation of the factorial function n! of a whole number n ∈ N. To extend the domain of deﬁnition of the zeta function, we ﬁrst note that the series for ζ (s) converges only when re z > 1. However, it is easy to verify that

´

1−

2 2s

µ

ζ (s) = 11s − 21s + 31s − · · ·

The series on the right converges for re z > 0, so the formula lets us extend the deﬁnition of the zeta function to such z, and it remains an analytic function of z. Then, starting from a formula of Adolf Hurwitz, Riemann deduced the functional equation for the zeta function: if 0 < re s < 1, then

¶ · ζ (s) = 2s π s−1 sin s2π ² (1 − s)ζ (1 − s)

where ² denotes the gamma function, Section 6.11. This formula can then be used to deﬁne ζ (s) for all complex z ± = 1, by assuming it holds whenever 0 ≤ re s, giving an analytic continuation of the zeta function to C \ {1}. The same formula shows that

( )

ζ (−2n) = 0

n = 1, 2, 3, . . .

because the factor sin s2π vanishes at these numbers. The negative even integers are the trivial zeros of the zeta function. In his paper, Riemann [16] observed that there also exist non-trivial zeros of the zeta function – those that are not negative even integers. In particular he found zeros at 1 2

² 14.135i

1 2

² 21.022i

1 2

² 25.011i

but did not publish these results. His calculations suggested that all non-trivial zeros seem to have real part 21 . This statement is now known as the Riemann Hypothesis. (His actual statement was about a closely related function and is equivalent to this.) He wrote:

17.2 Modular Functions

377

One would, however, wish for a strict proof of this; I have, though, after some ﬂeeting futile attempts, provisionally put aside the search for such, as it appears unnecessary for the next objective of my investigation.

As time passed, it became clear that the Riemann Hypothesis is of central importance in mathematics. It implies strong bounds on the error in the Prime Number Theorem – the order of magnitude of the difference between the approximate formula and π (x). For example, in 1901 Niels Helge von Koch showed that if the Riemann Hypothesis is correct then the error |Li(x) − π (x)| grows no faster than a constant times √ x log x, and in 1976 Lowell Schoenfeld gave the explicit bound

|Li(x) − π (x)| < 81π

x log x

(x ≥ 2657)

The Riemann Hypothesis also tells us about the size of gaps between consecutive primes. Better still, there are far-reaching generalisations with major implications throughout number theory, and it has applications to testing numbers to see whether they are prime, which is important for encrypted Internet communications. There is a lot of circumstantial evidence in favour of the Riemann Hypothesis, and computer calculations have veriﬁed it for the ﬁrst ten trillion zeros. Despite a huge amount of effort, however, the Riemann Hypothesis has neither been proved nor disproved. Hilbert included it in his famous list of unsolved problems in 1900. The Clay Mathematics Institute is currently offering a prize of one million dollars for solutions to six major unsolved problems in mathematics, and the Riemann Hypothesis is one of them. A seventh, the Poincaré Conjecture, was solved in 2002–3 by Grigori Perelman, who declined the prize.

17.2

Modular Functions Extending the idea of periodic functions (such as exp) leads to doubly periodic functions, which satisfy f (z) = f (z + p) = f (z + q) for two complex numbers p, q that are linearly independent over the reals (not real multiples of each other). These are also known as elliptic functions because some of them can be used to give a formula for the arc-length of an ellipse. Further generalisation leads to modular functions, which transform in a nice way under a discrete group of Möbius maps. Here ‘discrete’ means that each transformation is separate from the others rather than being part of some continuous family. For example, Z2 is a discrete subgroup of R2 . These functions preoccupied many mathematicians at the end of the nineteenth century. These united in one package group theory, differential equations, algebraic function theory, topology, and complex analysis. This is still an important ﬁeld of research. For example, modular functions are heavily involved in Andrew Wiles’s celebrated proof of Fermat’s Last Theorem. The starting point is the modular group, which consists of all Möbius maps

+b µ(z) = az cz + d

378

The Road Goes Ever On . . .

for which a, b, c, d ∈ Z and ad − bc ± = 0. This group is also called the special linear group over Z, denoted SL2(Z). If H denotes the upper half-plane {z ∈ C : im z ≥ 0}, then (subject to some technical conditions omitted here) a modular function is a function f : H → C that is analytic on H and satisﬁes the condition f (µ(z))

= (cz + d)k f (z) ∀z ∈ H

for any µ ∈ SL2 (Z), where the weight k is an odd positive integer. Generalisations of these functions, known as modular forms, have many applications to number theory. Wiles deduced Fermat’s Last Theorem from his proof of a longconjectured relationship between modular forms and certain number-theoretic equations called elliptic curves because of the connection with elliptic functions. An ellipse is not an elliptic curve.

17.3

Several Complex Variables Functions of several complex variables may be studied. They turn out to be far trickier than we might expect, with startling new phenomena. A function of several complex variables is a map f :D→C

where the domain D is an open subset of Cn for some n ∈ N, n ≥ 2. We require f to be analytic, in the sense that for each a = (a1 , . . . , an ) ∈ U there is a convergent power series expansion f (z1, . . . , zn)

=

±

bk 1,...,kn (z1 − a1 )k1 · · · (zn − an)kn

There is an equivalent condition analogous to the Cauchy–Riemann Equations, which can be proved using a form of Cauchy’s Integral Formula. A more common term in this context is that such an f is holomorphic on U. Some of the basic theorems and concepts of complex analysis generalise to several variables, but many others do not. Analytic continuation has a natural deﬁnition, and is again unique but possibly multivalued. One of the most inﬂuential new discoveries is the Hartogs’ phenomenon, named after Friedrich Hartogs. In one-variable complex analysis, there are analytic functions with a natural boundary – a closed curve beyond which they cannot be analytically continued. Equation (14.5) gives an example. Hartogs’ extension theorem of 1906 states, roughly speaking, that no such thing occurs for two or more variables. The domain on which a function is holomorphic is always unbounded. Hartogs proved more. In one-variable complex analysis, for any domain D there exists an analytic function f : D → C for which the boundary ∂ D is a natural boundary. Hartogs proved that if

³2 = {(z1, z2) ∈ C2 : |z1| < 1, |z2 | < 1} Hε = {(z1, z2) ∈ ³ 2 : |z1 | < ε or 1 − ε < |z2 |}

for 0 < ε < 1, then any function f holomorphic on Hε has a holomorphic extension F to ³2 . Indeed, F can be deﬁned using Cauchy’s Integral Formula.

17.5 Complex Dynamics

379

The theory was developed by Hartogs and Kiyoshi Oka in the 1930s. Hartogs proved that every isolated singularity is removable when n ≥ 2, another crucial difference from the one-variable case. The subject really began to take off in 1945 with work of Henri Cartan, Hans Grauert, and Reinhold Remmert. It soon evolved into the more general context of complex manifolds, which we consider next.

17.4

Complex Manifolds The Riemann surface alone has opened up broad vistas, by capturing the essential structure of a complex function in a single geometric object. All kinds of information, such as the presence of branch points and singularities, can be read off from it – and the ‘rigidity’ of analytic functions means that they are essentially determined by the position and nature of their singularities. Many questions that seem bafﬂing without Riemann surfaces become transparent when this extra geometry is invoked. The notion of a complex manifold arises when we generalise Riemann surfaces to several complex variables. The basic deﬁnition is modelled on real manifolds: these are multidimensional analogues of surfaces, obtained by patching together open subsets of Rn whose coordinates are smoothly related when the patches overlap. In the complex case we use open subsets of Cn whose coordinates are analytically related when the patches overlap. In differentiable topology a smooth real manifold may have several distinct differentiable structures, but in all dimensions except 4 the number of such structures is ﬁnite. However, a ﬁxed manifold of even dimension can possess inﬁnitely many different complex structures. These can be classiﬁed by associating each complex structure with a point in a so-called moduli space which is itself a complex algebraic variety – roughly, the set of zeros of a system of polynomials in several complex variables. The whole area is deeply related to topology and to algebraic geometry, and we will not attempt to summarise it. Because all of these areas are highly abstract, until recently much of this work was viewed by most non-mathematicians (among those aware of its existence) as mere generalisation for its own sake – pretty, intellectually clever, but far too wild and abstract to have sensible applications outside pure mathematics. Such judgements are usually superﬁcial and premature when they refer to the mathematical mainstream: most ideas that are important in pure mathematics eventually acquire signiﬁcant uses. And so it has proved here. Complex manifolds and automorphic functions are now central to the physics of Quantum Field Theory and the study of gauge ﬁelds in particle physics, for instance.

17.5

Complex Dynamics Let f : Z → Z be a map from a set Z to itself. Then we can apply f repeatedly to any z0 ∈ X to get the sequence z0, f (z0 ), f (f (z0)), f (f (f (z0 ))), . . .

This process is classically known as iterating f . It can be deﬁned inductively by

380

The Road Goes Ever On . . .

f 0 (z0)

= z0

f n+1(z0 ) = zn+1

= f (zn )

The modern term is discrete dynamical system. If we think of the subscript n as ‘time’, ticking in whole number steps, the sequence (zn) speciﬁes how an initial state x0 changes over time. Usually the set Z is a smooth manifold – a multidimensional analogue of a smooth surface. A central aim of dynamical systems theory is to understand sequences of this kind, especially their long-term behaviour. Complex dynamics studies discrete dynamical systems when Z = C (or some subset of C) and f is differentiable. The ﬁrst serious work on this topic was carried out by Pierre Fatou in 1917 and Gaston Julia in 1918. Their methods were analytic. With the advance of computer graphics it became possible to draw accurate pictures, which revealed surprisingly intricate shapes of great beauty. The associated mathematics is also very beautiful, from a logical point of view. However, many interesting and important problems remain open at this stage. For this brief discussion, we restrict attention to polynomial f . Analogous concepts can be studied for general differentiable f , but some deﬁnitions are more complicated. See Devaney [3] for the details. A key concept is the Julia set of f . This can be characterised in several equivalent ways, by proving appropriate theorems. Perhaps the simplest is to distinguish two types of behaviour for the sequence (zn). Depending on z0 , either the set of all zn remains bounded for all n ∈ N, or it diverges to inﬁnity. The Julia set J(f ) is the boundary of the set of z0 for which it diverges to inﬁnity. For polynomial f this is also the boundary of the set of z0 for which it remains bounded. The Julia set is invariant under f , that is, f −1(J(f )) = f (J(f )) = J(f ). For example, if f (z) = z2 then J(f ) is the unit circle. On J(f ), the map f sends eiθ to 2i e θ , doubling the angle. If f (z) = z2 − 2 it can be proved that J(f ) is the line segment [−2, 2], and the dynamics of f on this interval is more complicated. The geometry of Julia sets is fascinating, even for very simple maps f . It is best understood (and even then not completely) for quadratic maps f (z) = fc (z) = z2 + c

c∈C

Sometimes J(fc ) is quite simple. For instance, if c < 41 then J(fc ) is a simple closed curve. For other values of c it is a highly intricate fractal, as in Figure 17.1. For some c the Julia set is a disconnected dust cloud; for others it is connected.

Figure 17.1 Two examples of Julia sets.

17.6 Epilogue

381

Figure 17.2 Left: The Mandelbrot set. [From Wikimedia Commons under the GNU Free Documentation License.] Right: Close-up of the boundary of part of the Mandelbrot set.

To bring order to the Julia sets, we consider how their topology depends on the constant c ∈ C. The Mandelbrot set M ⊆ C is deﬁned to be the set of c for which J(fc ) is connected. Figure 17.2 (left) shows the Mandelbrot set: it is the black cactus-like region. The boundary of the Mandelbrot set is inﬁnitely complicated – a small part is shown in Figure 17.2 (right). Numerous images can be found on the Internet, including movies that zoom into the set at ever greater magniﬁcation. Near any point c the Mandelbrot set looks like the Julia set J(f c). It has been proved that the Mandelbrot set is connected. It is conjectured to be locally connected (look up the deﬁnition!) but this conjecture remains open as we write. Analogues for other types of differentiable complex function have been studied, and many problems remain open.

17.6

Epilogue In a sense, the wheel has come full circle. The contour of history has wound round the singularity of mathematical discovery, and closed. More precisely, it has returned to its starting point but climbed one level up the Riemann surface of scientiﬁc advances. In its early days, complex analysis was (almost) a branch of physics; the connections with potential theory and ﬂuid mechanics were widely exploited. Towards the end of the nineteenth century Felix Klein offered a ‘proof’ of a theorem along the following lines: think of the Riemann surface as being made of thin metal, and an electric current ﬂowing through it . . . . Today this would not be considered a logically convincing argument, but the physical intuition behind it led to some important mathematical discoveries. Now we are witnessing the converse process, with mathematical intuition leading to important ideas in physics. There is a two-way trade between mathematics and its applications. And whatever the attractions of beauty for its own sake, this trade is vital for the health of both mathematics and science.

References

[1] J. Anderson. Fundamentals of Aerodynamics (2nd edn), McGraw-Hill, Toronto 1991. [2] M. Bader. Space-Filling Curves, Texts in Computational Science and Engineering 9, Springer, Heidelberg 2013. [3] R. L. Devaney. Chaotic Dynamical Systems (2nd edn), Addison-Wesley, Redwood City 1989. [4] K. Falconer. Fractal Geometry, Wiley, Chichester 1990. [5] M. Flashman. Mapping Diagrams to Visualize Complex Analysis, 2015; www.geogebra. org/book/title/id/Ni69jyKs [6] M. Flashman. 2017; Retrieved from www.geogebra.org/o/bGUrTgZq. [7] J. Gray. Plato’s Ghost, Princeton University Press, Princeton 2008. [8] A. Hatcher. Algebraic Topology, Cambridge University Press, Cambridge 2009. [9] J. G. Hocking and G. S. Young. Topology, Addison-Wesley, Reading 1961. [10] H. J. Keisler. Foundations of Inﬁnitesimal Calculus, Prindle, Weber & Schmidt, Boston 1976. [11] M. Kline. Mathematical Thought from Ancient to Modern Times, Oxford University Press, Oxford 1972. [12] A. Kyrala. Applied Functions of a Complex Variable, Wiley, New York 1972. [13] B. Mandelbrot. The Fractal Geometry of Nature (2nd edn), Freeman, San Fancisco 1982. [14] T. Needham. Visual Complex Analysis, Oxford University Press, Oxford 1997. [15] R. Remmert. Theory of Complex Functions, Springer, New York 1998. [16] B. Riemann. ‘Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse’, Monatsberichte der Berliner Akademie (1859): 136–144. [17] A. Robinson. Non-standard Analysis, North Holland, Amsterdam 1966. [18] H. Sagan. Space-Filling Curves, Springer, New York 1994. Reissued 2013 in Universitext series. [19] I. Stewart. Why Beauty is Truth, Basic Books, New York 2007. [20] I. Stewart. Signiﬁcant Figures, Proﬁle, London 2017. [21] I. Stewart and D. O. Tall. Foundations of Mathematics (2nd edn), Oxford University Press, Oxford 2015. [22] K. D. Stroyan. ‘Uniform continuity and rates of growth of meromorphic functions’, in Contributions to Non-Standard Analysis (eds. W. J. Luxemburg and A. Robinson), NorthHolland, Amsterdam 1972, 47–64. [23] D. O. Tall. ‘Looking at graphs through inﬁnitesimal microscopes, windows and telescopes’, Mathematical Gazette 64 (1980): 22–49. [24] D. O. Tall and M. Katz. ‘A cognitive analysis of Cauchy’s conceptions of function, continuity, limit, and inﬁnitesimal, with implications for teaching the calculus’, Educational Studies in Mathematics 86 (2014): 97–124. [25] J. Wallis. A Treatise of Algebra, John Playford, London 1685.

Index

π , 100

analytic deﬁnition of, 100 series for, 110

absolute value, 17, 60 absolutely convergent, 65, 68, 69, 71, 209 acceleration, 9 aerofoil, 283 al-Khwarizmi, Muhammad, 1 Al-kitab al-mukhtasar ﬁ hisab al-gabr wa’l-muqabala of al-Khwarizmi, 1 Algebra of Wallis, 2 algebraic geometry, 379 algebraic topology, 350 algebraic variety, 379 alternating series, 65 amplitwist, 343 analytic, 212, 218, 289 at inﬁnity, 237 analytic continuation, 140, 289, 293, 295, 296, 298, 300, 346, 378 direct, 293–295, 298 indirect, 295 analytic function, 295–297 angle, 146, 149–151, 271 measurement of, 268 annulus, 43, 225, 226, 230 antiderivative, 91, 133–135, 143, 169, 170, 173, 175, 176, 183, 212, 213, 308 local, 176, 202, 203 antidifferentation, 112 approximating polygon, 117, 120 approximation, 374, 375 arc, 270 arc length, 126, 150, 191 Argand diagram, 4, 16 Argand, Jean-Robert, 4, 7, 13, 16, 341 argument, 20, 151, 163, 176 principal value of, 151 Arithmetic modulo n, 268 Arithmetica of Diophantus, 1 Ars Magna of Cardano, 1, 2 associative, 14 asymptotic, 375

asymptotic series, 224 automorphic function, 379 aviation, 283 axiom, 333, 334 of choice, 336, 342 of completeness, 334 bell curve, 257 Berkeley, George, 317 Bernoulli paradox, 165 Bernoulli, James, 59 Bernoulli, John, 3–5, 297 Bessel function, 73 Bessel, Friedrich, 4, 5 bilinear map, 278 Binomial Theorem, 70, 341 blancmange function, 87, 89, 124 Bolyai, Wolfgang, 5 Bolzano, Bernard, 35, 89 Bombelli, Raphael, 2, 7 boundary, 12, 191, 220, 350, 352, 353, 356, 358, 360 contour, 171, 177, 191 point, 220 bounded, 50, 51, 213 branch point, 298 calculus, 315, 316, 333 Cantor, Georg, 52, 336 Carathéodory, Constantin, 281 Cardano, Girolamo, 1, 2, 5 Cartan, Henri, 379 Cartesian coordinates, 19 Cauchy Integral Formula, 208, 209, 213, 227, 378 Cauchy principal value, 249, 251, 252, 254 Cauchy sequence, 62, 336 Cauchy Theorems compared, 202 Cauchy’s Estimate, 213, 214 Cauchy’s Residue Theorem, xi, xii, 243, 244, 246, 260, 261, 309, 370 homology version, 350, 368, 370 Cauchy’s Theorem, xi, xii, 5, 11, 12, 169–171, 177, 179–183, 187, 189, 190, 203, 204, 207, 243, 246, 308, 350, 353, 361 applications of, 180

384

Index

for a boundary, 189, 191, 194, 196, 200, 201, 351, 352, 362 for a triangle, 171, 177 for simple closed step path, 185 generalised version of, 181 homology version, 310, 350, 353 proof of, 362 homotopy version, 187, 350 Cauchy, Augustin-Louis, 5, 6, 12, 59, 79, 169, 208, 211, 212, 317 Cauchy–Riemann Equations, 5, 75, 78–82, 91, 93, 108, 282, 378 Cellérier, Charles, 89 chain, 352–354, 356 chain rule, 77, 83 fallacious proof, 78 Chebyshev, Pafnuty, 375 circle, 35, 42, 100, 125, 269, 278, 279 circle of convergence, 67 Clay Mathematics Institute, 377 closed contour, 170, 179, 181, 183 closed set, 26, 27, 51 closure, 27, 52 combined path, 40 commutative, 14 compact, 51 comparison test, 65 complement, 26, 50 of a path, 50 complete analytic function, 296, 301 complete ordered ﬁeld, 315, 331, 342 complex analysis, 1, 5, 7, 10, 24, 51, 52, 111, 374 applications, 6 special ﬂavour, 207 conjugate, 18, 19 dynamics, xii, 12, 379 function, 52, 59 manifold, xii, 12, 379 number, 1–5, 7, 10, 13 as a ﬁeld, 14 as a plane, 13, 16 construction of, 13 notation for, 15 plane, 35, 37 power, 302 complexity, 365 component, 49, 182 composition, 32, 77 computational ﬂuid dynamics, 283 computer graphics, 318 conformal, 268, 272, 277, 282, 283 function, 272 map, 272, 304, 305 mapping, 272 transformation, 268, 271, 272

congruent modulo 2π , 268, 269 connected, 24, 46, 82, 160, 183, 350, 381 connected component, 49 continuity, 24, 30, 32, 176 continuous, 30, 32–34, 76, 151, 152, 171, 191, 270 at a point, 30 at an isolated point, 31 continuous choice of argument, 155–158, 162 continuous complex function differentiable nowhere, 91 continuous deformation, 188 continuous real function differentiable nowhere, 89 contour, 130, 159, 170, 176, 188, 189, 298 closed, 131, 143 contour integral, 131, 305, 308 properties of, 131 contour integration, 130 converge, 63, 65, 225 convergence, 254 convergent, 60, 63, 65 cos + i sin formula, 70, 96 cosine, 59, 69, 99, 150 Cotes, Roger, 4, 7 critical point, 128, 276, 320 cross ratio, 287 cubic equation, 2 curve, 37 regular, 125 cusp, 127, 128 cut, 182, 203, 227, 308 cut plane, 151, 152, 154, 155, 163, 175, 176, 308 cycle, 12, 352, 353, 356 d’Alembert, Jean le Rond, 4, 79 de la Vallée-Poussin, Charles Jean, 376 De Moivre’s Formula, 99 De Moivre’s Theorem, 22, 23 de Moivre, Abraham, 7, 59 deﬁnite integral, 111 del Ferro, Scipione, 1 dense, 53 derivative, 75, 171, 316 Descartes, René, 2 differentiable, 75–77, 81, 82, 84, 92, 145, 171, 175–177, 179, 183, 189, 212, 213, 236, 243, 244, 268 at inﬁnity, 237 implies n times differentiable, 75 differential equation explanation of Euler’s formula, 103 differentiation, 24, 75, 76, 83, 111 rules for, 76 term by term, 84, 85 Diophantus of Alexandria, 1 direction preserving, 39

Index

same, 39 Dirichlet series, 374 Dirichlet’s Discontinuous Factor, 266 Dirichlet, Johan Peter Gustav Lejeune, 266, 374 disc, 218, 225, 289, 295 disc of convergence, 67, 84, 86, 292 discovery, 8 discrete, 377 discrete dynamical system, 380 distributive, 14 divergent, 63 division point, 112 Dog Walking Theorem, 371 domain, 24, 51, 52, 75, 82, 152, 169, 170, 177, 179, 183, 218, 244, 297, 350 doubly periodic function, 377 dyadic rational, 91, 124 e, 97 ellipse, 377 elliptic curve, 378 elliptic function, 377 elliptic modular function, 234 end point, 35 epsilon-delta, 91, 315, 316, 333, 342 equation cubic, 1 quartic, 1 equipotential line, 282 Estimation Lemma, 140, 142, 144, 160, 174, 208, 210, 260 Euclid, 9 Euclidean geometry, 268, 278 Euler’s constant, 139 Euler’s formula, 3, 4, 96, 99, 104 Euler, Leonhard, 3, 5, 7, 9, 15, 59, 103, 137, 139, 260, 297, 341, 375 zeta function formula, 375 exponential function, 11, 20, 59, 69, 96, 104, 149, 153 periodicity of, 104 real, 98 extend, 217 extended complex plane, 234, 236–238 extended number line, 315 extension function, 217, 218 Fatou, Pierre, 380 Felkel, Anton, 374 Fermat’s Last Theorem, 377, 378 Ferrari, Lodovico, 1 ﬁeld, 14, 16 ﬁeld of view, 332 ﬁnal point, 35, 40 ﬁnite, 325–327 complex, 331 element, 315

385

ﬁrst homology group, 353 ﬁrst order logic, 334 ﬂuent, 341 ﬂuid dynamics, 282, 304 ﬂuxion, 341 Fontana, Niccolo (‘Tartaglia’), 1, 2 force, 9 fractal, 90, 124, 380 Frankenstein, 88 Frankenstein’s monster, 88 free abelian group, 355 function element, 296 functional equation for the zeta function, 376 fundamental group, 205 Fundamental Theorem of Algebra, 4, 214, 263 Fundamental Theorem of Calculus, 112 Fundamental Theorem of Contour Integration, 133, 134, 138, 169, 212 Galileo, Galilei, 8 gamma function, 137 basic properties, 138 derivative, 139 duplication formula, 139 functional equation, 138 reﬂection formula, 139 gauge ﬁeld, 379 Gauss plane, 16 Gauss, Carl Friedrich, 4, 5, 7, 13, 15, 16, 169, 341, 374 gaussian, 257 General Principle of Convergence, 62 Generalised Cauchy Theorem, 208, 245 GeoGebra, 345, 346 ghosts of departed quantities, 317 graph, 128, 319 Grauert, Hans, 379 Greeks, ancient, 124 Gregory’s series, 59, 110 Gregory, James, 211 grid point, 363 Hadamard, Jacques, 376 Hahn, Hans, 55 Hahn–Mazurkiewicz Theorem, 55 halo, 330 Hamilton, William Rowan, 5, 7, 13 Hardy, Godfrey Harold, 89 harmonic conjugate, 282 function, 282 series, 118, 130 Hartogs phenomenon, 378 Hartogs’ extension theorem, 378 Hartogs, Friedrich, 378, 379 Hausdorff topological space, 55 higher derivative, 213

386

Index

higher homology group, 353 higher order, 329 complex, 331 Hilbert, David, 52, 377 hole, 187 holomorphic, 378 homologous, 352, 360 homologous to zero, 352 homology, xi, xii, 12, 187, 204, 310, 350–353, 357, 360, 361, 364, 366 class, 353 group, 12, 360 homotopic to zero, 200–203 homotopy, xii, 187, 188, 195, 196, 201–203, 350 closed path, 196, 198, 200 ﬁxed end point, 196–198, 201 Hopf, Heinz, 353 horse joke, 64 human eye, 315, 344 Hurwitz, Adolf, 376 hybrid function, 75, 83, 128 hyperbolic function, 106 hypercomplex numbers, 337 hyperreal numbers, 333–335, 337 construction of, 336 ideal, 328, 331 identity additive, 14 multiplicative, 14 Identity Theorem, 216, 217 image of cycle, 355 imaginary axis, 16 imaginary number, 2, 5 imaginary part, 2, 17, 29, 30, 65, 75, 133 indentation, 254 Indian mathematics, 59 inﬁnite, 326, 327 complex, 331 inﬁnite element, 316 inﬁnite of order k, 325 inﬁnite series, 258 inﬁnitely differentiable function not equal to Taylor series, 88 inﬁnitesimal, xi, 6, 12, 315–318, 322–324, 326–333, 337–339, 341–344 complex, 331, 332 how to visualise, 328, 329 in education, 342 of ﬁrst order, 316 of order n, 325 of second order, 316 ridiculed, 317 inﬁnitesimal δ -shape, 337, 338 inﬁnity, 235 behaviour of differentiable function at, 236

initial point, 35, 40 inside, 182, 183, 243, 244 integers made by God, 1 integral, 122, 169, 176, 189, 298 additivity of, 355 along arbitrary path, 189 calculation of, 180, 246, 248 Cauchy principal value of, 249, 251, 252, 254 depending on path, 136 evaluation of, 3, 11 over a chain, 352, 355 integration, 24, 111, 112, 243 by parts, 147 integration by parts, 138 Intermediate Value Theorem, 98, 100 In the Neolithic Age of Kipling, 202 invention, 8 inverse additive, 14 cosine, 153 function, 149 multiplicative, 14 sine, 152 trigonometric function, 152 inversion, 280 isolated essential singularity, 231, 234, 298 at inﬁnity, 236, 237 isolated point, 27, 31 isolated singularity, 230–232, 236, 238, 243, 244 iteration, 379 Jordan contour, 181–183, 243 Jordan Contour Theorem, 183, 371 for step path, 185 Jordan’s inequality, 265 Joukowski aerofoil, 283 transformation, 283 Julia set, xii, 12, 380, 381 Julia, Gaston, 380 Jupiter, 8 Jyesthadeva, 59 Karmann–Trefftz transform, 283 Keisler, Howard Jerome, 343 Kerala school of mathematics, 59 Kipling, Joseph Rudyard, 202, 203 Klein, Felix, 381 Kline, Morris, 7 Kronecker, Leopold, 1 La Géometrie of Descartes, 2 Lagrange, Joseph-Louis, 211 Laplace equation, 281–283 Laplace transform, 265 Laurent expansion, 229, 231, 232, 243, 245

Index

uniqueness of, 229 Laurent series, 6, 11, 229, 231, 234, 245, 247, 291 Laurent’s Theorem, 226 Laurent, Pierre-Alphonse, 6, 225 left derivative, 88 Legendre, Adrien-Marie, 374 Leibniz, Gottfried Wilhelm, xii, 3–5, 13, 297, 316, 322–324, 326, 329, 341 Leibniz–Bernoulli controversy, 3–5 length, 39, 117, 123 lens, 328–330, 344 complex, 332 limit, 24, 27, 28, 30, 60, 75, 77 reduction to real case, 61 limit point, 27, 31, 215, 217 line, 125 line segment, 35, 36, 42 Liouville’s Theorem, 213, 214 Liouville, Joseph, 213 local maximum, 219, 221 on subset, 220 strict, 219 local minimum, 219–221 strict, 219, 220 logarithm, 3, 4, 11, 96, 137, 149, 175, 176, 298, 299, 303 complex, 4, 153 derivative of, 154 of complex number, 149 of negative number, 149 principal value of, 154, 175, 298 real, 98 logarithmic integral, 374 loop simple, 243 lower order, 329 complex, 331 Mathematica, 345 Möbius map, 268, 278–281, 377 composing, 278 maps circles to circles, 279, 280 Madhava of Sangamagra, 59 magniﬁcation, 315, 318 magnify, 327 Mandelbrot set, xii, 12, 381 manifold, 378–380 mathematical logic, 333 Maximum Modulus Theorem, 220, 221 Mayer, Walther, 353 Mazurkiewicz, Stefan, 55 Mean Value Theorem, 39, 80 meromorphic, 237–239 microscope, 328–330, 332, 343 Minimum Modulus Theorem, 221 modular form, 378

387

function, xii, 12, 377, 378 group, 377 moduli space, 379 modulus, 17, 65, 219 monad, 330, 334, 337, 341 complex, 332 Moore, Eliakim Hastings, 171 Morera’s Theorem, 212 Morera, Giacinto, 212 multiform, 296, 298–300, 302, 304, 305, 308, 310 function, 296, 297 multivalued, 149, 154 natural boundary, 298, 378 natural logarithm, 98 Needham, Tristan, 343 negative inﬁnite, 326 negative inﬁnitesimal, 326 neighbourhood, 26 Netto, Eugen, 52 Newton, Isaac, 2, 9, 59, 211, 316, 341 Noether, Amalie Emmy, 353 non-standard analysis, 326, 333, 335, 342 non-winding condition, 362 North pole, 235 null-homotopic, 200 Oka, Kyoshi, 379 Oldenburg, Henry, 59 one-point-compactiﬁcation, 235 open, 350 open set, 24, 26, 51, 82 opposite path, 42 optical lens, 332, 337 optical microscope, 333, 337–339 order of inﬁnitesimality, 325, 329, 341 ordered ﬁeld, 324, 326, 334 ordering does not exist for complex numbers, 21 of real numbers, 20 Osborne’s rule, 108 outside, 182, 183 parameter, 37 change of, 38, 39 parametric interval, 37, 39, 41 parametrisation, 37 smoothly equivalent, 123 Parseval’s Inequality, 223 partial derivative, 79, 81 fraction, 3 sum, 63 particle physics, 379 partition, 112, 189, 190 path, 24, 35, 40, 48, 111, 150, 155, 169, 189, 298 complement of, 160

388

Index

component of, 160 in a subset, 43 length of, 122 of inﬁnite length, 118 piecewise smooth, 112 regular, 9, 125, 127 smooth, 111, 119, 122, 124, 125 integral along, 113 length of, 117, 119 path-connected, 24, 47–49, 51, 350 component, 49 pavable, 46, 194 Paving Lemma, 24, 43, 45, 46, 49, 156, 170, 189, 190 Peano, Giuseppe, 52 Perelman, Grigori, 377 period, 104 periodicity, 149, 151, 377 pi function, 148 Picard’s Theorem, 234 Picard, Émile, 234 Platonism, 10 Poincaré Conjecture, 377 polar coordinates, 19, 70 pole, 231, 232, 234, 237, 238, 250, 292, 298, 299 at inﬁnity, 237 of order m, 231, 233, 247 at inﬁnity, 236 on real axis, 254 simple, 246, 258 polygon, 54, 189, 190, 316 approximating, 117, 123, 124 polygonal approximation, 190 polynomial, 34, 84, 214, 238, 239 derivative of, 84 positive inﬁnite, 326 positive inﬁnitesimal, 326 potential function, 282 potential theory, 281, 282 power, 155 principal value of, 155 power series, 11, 59, 66, 75, 84, 92, 99, 150, 151, 207, 209, 211, 212, 217, 225, 289, 322, 326, 333, 337, 378 comparing, 291 convergence of, 66 limitations of, 289 manipulation of, 69 prime, 374, 375 prime factorisation unique, 375 Prime Number Theorem, 375–377 punctuation, xiii punctured plane, 369 quadratic map, 380 Quantum Field Theory, 379

radian, 146, 150, 151 radius of convergence, 67, 68, 218 ratio test, 66 rational function, 238, 239, 324 ray, 153 real analysis, 51, 59, 69, 149 number, 2, 13, 315–317, 323, 324, 326–328, 330, 331, 333, 334, 336, 342, 344 real axis, 16 real part, 17, 29, 30, 65, 75, 87, 133 rectangle, 177, 191, 353, 363 in a domain, 352 relevant, 178, 179, 365 rectiﬁable, 124 regular, 9 regular point, 124, 127, 276, 320, 322 regular polygon, 7 relatively open set, 31 Remmert, Reinhold, 379 removable singularity, 231–234, 298 at inﬁnity, 236, 237 reparametrisation, 39, 126, 200–202, 358, 359 orientation-preserving, 359 residue, 6, 11, 243, 246, 247, 250, 258, 309 calculation of, 246 Riemann Hypothesis, xii, 12, 137, 140, 374, 376, 377 Riemann integral, 112, 113, 189, 333 complex, 114 Riemann sphere, 234, 236, 281 Riemann sum, 111, 112 Riemann surface, 6, 289, 296, 299–305, 308, 310, 340, 341, 346, 379, 381 Riemann, Georg Bernhard, 6, 12, 79, 140, 235, 299, 375, 376 Riemann–Stieltjes integral, 113 right derivative, 88 right triangle, 151 rigid motion, 268 ring, 328, 331 Robinson, Abraham, 333, 334, 342, 343 Rouché’s Theorem, 262, 263, 371 same order, 329 complex, 331 Schoenfeld, Lowell, 377 Schwartz Reﬂection Principle, 314 second derivative, 75 second order logic, 334 segment, 364 semicubical parabola, 129 sequence, 59 series involving negative powers, 225 summation of, 258 series involving negative powers, 225

Index

several complex variables, xii, 12, 378 sheet, 299, 300 simple loop, 243, 244, 246, 261 simply connected, 183 sine, 59, 69, 99, 150 singular point, 124, 127, 276 singularity, 218, 237, 244, 292, 295, 298, 300 smooth, 123 snowﬂake curve, 124 space-ﬁlling curve, 52, 91, 190 construction of, 52 special linear group, 378 spiral, 130 square root, 149 standard part, 327, 330 complex, 331 properties, 331 properties of, 328 star centre, 173 star domain, 170, 173, 175, 176, 193 stationary point, 128 step path, 48, 170, 176, 178–180, 350 step-connected, 48, 49, 51 streamline, 282, 283, 304 strictly increasing, 39, 126 Stroyan, Keith, 343 structure theorem, 315, 323, 343 for ordered extension ﬁeld of R, 327 for super complex ﬁeld, 331 subpath, 40 sum of paths, 40, 41 summation of series, 258 super complex ﬁeld, 331 super ordered ﬁeld, 326, 327, 330, 331, 334 not complete, 330, 331 supercomplex ﬁeld, 326 number, 333 superreal ﬁeld, 324, 326 number, 325, 326, 333 Tall, David, 343 tangent, 124, 316, 318 to curve, 125 Tartaglia, 1 Tartaglia’s formula, 2 Taylor coefﬁcient, 209, 215 Taylor expansion, 211, 217, 218, 225, 228 Taylor series, 87, 88, 91, 92, 207, 209, 211, 212, 214, 218, 225, 228, 231, 238, 290, 291, 338, 342 Taylor’s Theorem, 87, 207 Taylor, Brooke, 211 telescope, 328, 332, 343

389

tends, 60 term by term differentiation, 84 term of sequence, 60 top right point, 365 topology, 11, 12, 24, 46, 51, 52, 150, 187, 188, 268, 350, 379 transfer principle, 334, 336, 337 Trefftz, Erich, 283 triangle, 170 triangle inequality, 17 trigonometric function, 11, 59, 69, 96, 99–102, 105, 149, 150 periodicity of, 104 trigonometry, 96 ultraﬁlter, 336, 342 unbounded, 50 uniform, 296, 299 unit circle, 36 unpavable, 46, 194 upper half-plane, 378 Vandermonde, Alexandre-Théophile, 7 Vega, Jurij, 374 Vietoris, Leopold, 353 visual cortex, 9 von Kármán, Theodore, 283 von Koch, Niels Helge, 377 Wallis, John, 2, 7, 13 wedge, 304 Weierstrass, Karl, 6, 35, 59, 89, 139, 225, 289, 291, 315 Weierstrass–Casorati Theorem, 234 weight, 378 Wessel, Caspar, 4, 7, 13, 16, 341 Wiles, Andrew, 377, 378 winding number, 149, 150, 155, 158–160, 163, 169, 179, 180, 182, 187, 188, 353, 356, 360, 361 additivity of, 158, 159, 356 as homology invariant, 362 as integral, 159 computation by eye, 161 of a chain, 353, 355 to deﬁne homology, 361 Yuktibhasa of Jyesthadeva, 59 zero, 214 counting, 261 isolated, 215 of ﬁnite order, 215 of order m, 215, 273 at inﬁnity, 237 zeta function, 375 Zhukovsky, Nikolai, 283